Price Competition in an Oligopoly Market with Multiple IaaS Cloud Providers

Transcription

1 Prce Competton n an Olgopoly Market wth Multple IaaS Cloud Provders Yuan Feng, Baochun L, Bo L Department of Computng, Hong Kong Polytechnc Unversty Department of Electrcal and Computer Engneerng, Unversty of Toronto Department of Computer Scence, Hong Kong Unversty of Scence and Technology Abstract As an ncreasng number of Infrastructure-as-a- Servce (IaaS) cloud provders start to provde cloud computng servces, they form a competton market to compete for users of these servces. Due to dfferent resource capactes and servce workloads, users may observe dfferent fnshng tmes for ther cloud computng tasks and experence dfferent levels of servce qualtes as a result. In order to compete for cloud users, t s crtcally mportant for each cloud servce provder to select an optmal prce that best corresponds to ther servce qualtes, yet remanng attractve to cloud users. To acheve ths goal, the the underlyng ratonale and characterstcs n ths competton market need to be better understood. In ths paper, we present an n-depth game theoretc study of such a competton market wth multple competng IaaS cloud provders. We characterze the nature of non-cooperatve competton n an IaaS cloud market, wth a goal of capturng how each IaaS cloud provder wll select ts optmal prces to compete wth the others. Our analyses lead to suffcent condtons for the exstence of a Nash equlbrum, and we characterze the equlbrum analytcally n specal cases. Based on our analyses, we propose teratve algorthms for IaaS cloud provders to compute equlbrum prces, whch converge quckly n our study. Index Terms Cloud computng, Infrastructure-as-a-Servce, market competton, cloud prcng I. INTRODUCTION Cloud computng has recently emerged as a new paradgm for a cloud provder to host and delver computng servces to enterprses and consumers who use such servces. One of the possble types of cloud servces provded by today s cloud provders, such as Amazon EC2 and Rackspace, s referred to as Infrastructure as a Servce (IaaS). Wth IaaS, each physcal machne that a cloud provder hosts s vrtualzed usng a hypervsor, such as Xen Server. Such vrtualzaton makes t feasble for each physcal machne to host multple vrtual machnes (VMs), and computng resources are leased to cloud users n the form of these VMs. By mgratng from tradtonal n-house server nfrastructures to cloud computng, cloud users may trade a sgnfcant amount of up-front nvestment costs to the ongong costs of usng resources provsoned on-demand by IaaS cloud provders. In return, IaaS cloud provders are able to charge ther users for usng computng resources on a pay-as-you-go bass. Wth multple IaaS cloud provders avalable to the cloud users, one of the ways they may dfferentate themselves wth s ther pay-as-you-go prces of usng VMs for an hour, and such prces reflect the qualty of ther servces. For example, as of March 23, for each Vrtual Machne (VM) wth 4 CPU cores, Amazon EC2 charges $.24 for an hour of usage (called large on-demand nstance) [], and GoGrd charges $.32 [2], and Rackspace charges $.48 [3]. Snce a user s cloud servce demand may be satsfed by any of these IaaS cloud provders, a ratonal user wll choose the one that maxmzes ts own net reward,.e., ts utlty obtaned by choosng the IaaS cloud servce mnus ts payment. The utlty of a user s not only determned by the mportance of the task (.e., how much beneft the user can receve by fnshng ths task), but also closely related to the urgency of the task (.e., how quckly t can be fnshed). The same task, such as runnng an onlne voce recognton algorthm, s able to generate more utlty for a cloud user f t can be completed wthn a shorter perod of tme n the cloud. Snce the dversty among IaaS cloud provders wll lead to dfferent net rewards, multple IaaS provders form a market to compete for cloud users. Exstng real-world measurement results [4] reveal that dfferent IaaS provders complete tasks wth dfferent completon tmes, and an IaaS provder can become less compettve wth an napproprate prce settng. Wth dfferent prce settngs, payments made to fnsh each benchmarkng task are also dfferent across dfferent provders. As a consequence, the IaaS cloud provders are presented wth a queston: how can each provder compute the optmal prce to maxmze ts proft n such a competton market, n whch demands from cloud users are senstve to both the fnshng tme and the payment of completng a task? It turns out that answerng ths queston s non-trval. On one hand, IaaS cloud provders may wsh to ncrease the prce to generate more proft. On the other hand, ncreasng the prce too much n a compettve envronment may rsk losng potental cloud users, whch then results n a reduced amount of proft. Further, although reducng the prce should ntutvely be an effectve way to attract cloud users, these users may overwhelm the IaaS cloud provder due to an unreasonably low prce, whch then leads to longer fnshng tmes on the tasks to be completed. As a consequence, the reduced utlty wll prohbt future users to choose ths cloud provder. In ths paper, we take the frst step to study the prce competton n a cloud market formed by multple IaaS cloud provders. More specfcally, we present an n-depth analytcal study on the monopoly, duopoly and olgopoly markets, n whch multple IaaS cloud provders are competng wth one another. We use an M/M/ queue to model correlatons among the expected task fnshng tmes, an IaaS cloud provder s

2 2 resource capacty, and the request rates from cloud users. Snce the prcng strategy of a cloud provder depends on ts compettors, we take a game-theoretc perspectve to study the strategc stuaton. To our knowledge, ths s the frst study that dscusses the competton among IaaS cloud provders n the context of olgopoly market competton. Our orgnal contrbutons n ths paper hnge upon the suffcent condtons we have derved for the exstence of a Nash equlbrum n the market. By analyzng the Nash equlbrum, we make the followng observatons. Frst, when multple IaaS cloud provders compete for users, the cloud provder wth a larger resource capacty s able to charge a hgher prce and take more cloud users n equlbrum. However, ts proft wll not monotoncally ncrease wth larger resource capactes, due to ncreasng operatng costs. As a result, though ncreasng the resource capacty s an effectve way for a cloud provder to become more compettve n the market, t can only ncrease ts expected proft to a certan extent. If we take servcelevel obectves, securty measures, reputaton and brand nto consderaton, ncreasng the capacty of a datacenter may become even less effectve. Second, the equlbrum prce s found to be senstve to the mportance as well as the urgency of tasks of cloud users: t decreases wth the mportance and ncreases wth the urgency of tasks. Ths motvates the use of servce-level obectves for cloud users to further specfy the mportance and urgency of ther tasks. Thrd, the equlbrum prces are not always socally optmal. Fnally, we propose teratve algorthms to fnd equlbrum prces n the duopoly and olgopoly markets, respectvely, both of whch are shown to be convergng rapdly to the equlbrum. The remander of ths paper s organzed as follows. We show the orgnalty of our work n the context of related work n Sec. II. In Sec. III, we frst formulate the competton market and present our model, and then begn our analyss wth the monopoly problem, whch serves as the baselne for our comparsons. In Sec. IV, we analyze the competton between two IaaS cloud provders wth heterogeneous users, and propose an teratve algorthm to fnd equlbrum prces. We also study the correspondng socal welfare problem. We extend our dscusson to an olgopoly market n Sec. IV-B, and propose an algorthm to fnd Nash equlbrum prces for each cloud provder. Sec. V shows some characterstcs of Nash equlbrum prces wth extensve smulatons. In Sec. VI, we conclude the paper wth extensve dscussons on other mportant factors that nfluence the prcng strateges n a cloud market. II. RELATED WORK Consderable performance dfferences across cloud provders have attracted a substantal amount of research attenton. Hong et al. [5] and Tsakalozos et al. [6] appled dynamc programmng and mcroeconomcs, respectvely, to acheve optmal resource allocaton for cloud users n VM-based IaaS clouds, wth full awareness of dfferent prces charged by cloud provders. Exstng papers were concerned wth the problem of how optmal prcng n the cloud can be acheved. To fnd the optmal prce for a cachng servce n the cloud, Kantere et al. [7] modeled the correlaton between user demand and the prce, and proposed a dynamc prcng scheme to maxmze the cloud provder s proft. Teng et al. [8] and Mhalescu et al. [9] studed optmal prcng wth an aucton mechansm, n whch users had budgetary and deadlne constrants. Our prevous work [] consdered an exchange-based market for VMs, and proposed a soluton based on Nash barganng games. Xu et al. [] used a revenue management framework to maxmze a cloud provder s revenue wth dynamc cloud prcng. Our work n ths paper dffers substantally from prevous papers. Frst, all prevous works consdered the prcng of one provder alone, but our focus n ths paper s how optmal prcng can be determned n a compettve envronment wth more than one cloud provder. Second, most prevous models assume that the prce s a certan functon of user demand, whch has not been valdated n measurement studes. In contrast, we make the more realstc assumpton that user demand at each cloud provder remans unknown, and s subect to a game-theoretc analyss n a duopoly or olgopoly cloud market. Prce competton has been an actve research topc n the context of economc markets wth multple servce provders. Petr et al. [2], [3] have studed the effects of rsk n servcelevel agreements (SLAs) n servce provder communtes. Chen et al. have presented an analyss of the equlbrum prce n a monopoly market [4], and they have also dscussed equlbrum prces n a duopoly market wth varyng demand [5]. Allon et al. examned the scenaro that multple provders competed for users usng dfferent prces and tme guarantees [6]. The competton game among multple resource provders was also consdered n networkng research. Anselm et al. studed a congeston game wth multple lnks, each of whch was under the control of a proft maxmzng provder [7]. In the context of processor sharng queues, they dscussed the exstence and effcency of olgopolstc equlbra. Smlar to these exstng works, we are also nterested n the exstence of Nash equlbra n the cloud market wth multple IaaS cloud provders. Yet, the context of our study s prce competton n a cloud computng envronment, whch has a dfferent system model. In our model, each cloud user s assocated wth a dfferent request rate as t s served by the cloud, and such heterogenety n per-user request rates makes our analyses much more challengng. III. MODEL FORMULATION AND MONOPOLY ANALYSIS To begn wth, we present our system model n the context of IaaS cloud provders, and establsh mportant results wth respect to monopoly prcng, whch, whle beng the most elementary n our analyses, provdes us wth a sold understandng towards our man analytcal results that follow. A. System Model In ths paper, we are concerned wth a market wth multple IaaS cloud provders, who are competng for cloud users. Each cloud provder s modeled by an M/M/ queue, servng a common pool of potental cloud users wth one super server, whch combnes the resource capacty of multple physcal

3 3 Notaton TABLE I DEFINITIONS OF MATHEMATICAL NOTATIONS Defnton µ the servce rate of the cloud provder γ (µ ) the operatng cost at the cloud provder p the usage prce per VM at the cloud provder f the market share of the cloud provder v the reservaton value at cloud users M the number of cloud users λ the request rate at cloud user U (λ ) the utlty of cloud user by choosng to be served by the IaaS provder r the beneft factor per VM requested c the cost factor per tme unt π the expected proft of cloud provder Λ the market sze P the total payment user makes to cloud provder L the combned effects of other compettve factors t the expected fnshng tme of a unt request experenced at the cloud provder β the combned attracton of provder s compettors servers that the provder manages. When t comes to analyzng the response tme exhbted when processng requests as a functon of the computatonal capacty and the request arrval rate, the M/M/ queung model has been adopted by a number of exstng papers n the lterature that analyzed datacenter operatons [8] [2]. The resource capacty of each cloud provder s represented by ts servce rate µ. γ (µ ) s used to denote the operatng cost at cloud provder, whch s assumed to be a functon of ts resource capacty. For users who would lke to choose the cloud servce, the IaaS cloud provder wll charge a fxed per-tme-unt usage prce for each type of resources consumed to fnsh ther tasks. All operatonal IaaS cloud provders support on-demand prcng for users to use cloud computng resources. Ondemand prcng allows users to pay for the amount of resources consumed to complete ther tasks wth no long-term commtments. Wth ths prcng scheme, cloud provders charge users based on the amount of resources consumed to complete ther tasks. As a result, we use p r to denote the fxed usage prce per resource unt for example, an unt of CPU tme when usng a vrtual machne at an IaaS provder for a type of resource r. As we wll focus on the prce competton among multple IaaS provders for a gven type of resource, the ndces r wll be dropped for smplcty. The arrval of requests from cloud users are assumed to follow a Posson process, an assumpton that s commonly used n competton models n the economc lterature [4] [6]. A cloud user makes a choce to be served by a specfc cloud provder. Yet, t also mantans a reservaton value v (assumed to be the same across all users), and f by usng the cloud servce ts net reward falls below v, user can refuse to use any cloud servce, and choose to fnsh ts task locally. As shown n Fg., a user has a task wth requests for resources that t wshes to fnsh n the cloud. The rate at whch these requests are generated when runnng the task at a cloud provder s denoted by λ. The market share of a cloud provder s denoted by f, whch equals the sum of request rates of all users who choose cloud provder. Each cloud user Cloud Provders Cloud Provder Server Capacty: µ Operatng Cost: (µ )... Cloud Provder Server Capacty: µ Operatng Cost: (µ ) Usage Prce: p f Usage Prce: p f Cloud Users Cloud User Reservaton Value: v Utlty: U( ) Cloud User 2 Reservaton Value: v Utlty: U( 2)... Cloud User Reservaton Value: v Utlty: U( ) Fg.. Our model of competton n an olgopoly cloud market wth multple IaaS cloud provders. only selects one of the IaaS cloud provders,.e., t does not splt ts requests by routng them to multple IaaS provders smultaneously. Snce the cloud provder s modeled as an M/M/ queue wth a servce rate µ, based on queueng theory [22], the expected fnshng tme experenced by a request from one of the cloud users (called response tme n the queueng theory lterature), ncludng both the tme watng n the queue and the tme beng served, s µ f. Ths does not depend on the schedulng dscplne as long as t s work conservng, and can be computed usng Lttle s Law. It then follows that the expected tme that a user request spent watng n the queue s µ f µ, the second term beng the expected servce tme. Wth a request rate of λ at cloud provder, the expected fnshng tme s, therefore: + λ f = µ f µ µ µ (µ f ) + λ. () µ We focus on statonary analyss only, n that for a gven µ, the equlbrum market share f < µ must hold, snce otherwse the queue wll grow nfntely long and the system wll not have a statonary dstrbuton. For the problem to be worth dscussng, we assume f < µ throughout the entre paper. Now let us consder user s utlty, by choosng to be served by a cloud provder. Snce a user wll be more satsfed wth more tasks completed or faster servce n the cloud, ts utlty should become hgher as the request rate λ ncreases, and as the expected task fnshng tme becomes shorter. More formally, wth a request rate of λ, user s utlty s: [ U (λ ) = r λ c f µ (µ f ) + λ µ 2 ], (2) where the beneft factor r (per unt of resource requested) reflects the relatve mportance of the task, and the watng cost factor c (per tme unt) reflects ts urgency. The more mportant a task s, the more utlty s brought to the user. If a task s more urgent,.e., assocated wth a larger c, the tme consumed to complete the task ncurs a hgher watng cost. In Fg., t can be observed that multple cloud users wth dstnct request rates and utltes are makng a choce of ther respectve IaaS cloud provders, based on ther own utltes and on the prces charged by each cloud provder. Each IaaS cloud provder sets an approprate usage prce,

4 4 accordng to ther resource capactes and the prces from the other provders. The obectve of each cloud provder s to maxmze ts own expected proft, wth full awareness of possble reactons of other cloud provders and all the cloud users. Important notatons that we ntroduced so far are presented n Table I. Remark. In some exstng papers on the topc of analyzng cloud computng performance, e.g., [23] [25], a cloud provder s modeled as a M/M/c queue. Ths model s apparently motvated n the lterature by the fact that an IaaS cloud provder operates multple physcal servers, and each physcal server can be vewed as a server from a queueng theoretc perspectve n the M/M/c model. If we adopt ths model, the resource capacty of each cloud provder can then be represented by ts servce rate µ = n µ, where the cloud provder has n servers, each of whch has a servce rate of µ Ḃased on the theory of M/M/c queues [22], the expected fnshng tme experenced by a user wth a request rate of p q n µ f µ λ at cloud provder s + λ µ, where p q s the probablty that an arrvng request from a user wll be forced to on and wat n the queue (all servers are occuped), and s gven by C(n, f /µ ), called Erlang s C formula. Due to the large number of physcal servers n a typcal IaaS cloud provder, we beleve that t s suffcent to abstract multple servers as a sngle super server to serve the market n the context of cloud provders, and the addtonal modelng complexty s not necessary. B. An Analyss of Monopoly Prcng We are now ready to present our analytcal results on monopoly prcng, whch serve as prelmnares and a bass for later comparsons. In ths subsecton, we consder a sngle cloud provder modeled by an M/M/ queue, wth a servce rate µ and an operatng cost γ(µ). A ratonal cloud user wll seek to maxmze ts expected net reward by fnshng the task,.e., ts utlty obtaned by choosng the cloud servce mnus ts total payment. Snce cloud users are charged based on how much resource they consume, cloud user s total payment P = pλ,. Now that there s only one cloud provder n the market, ths mples that the cloud user wll choose to use the cloud servce f U(λ ) P v and refuse to use t otherwse. In equlbrum, U(λ ) P = v, wth the correspondng market share f < µ. In equlbrum, ths mples: [ rλ c f µ(µ f) + λ µ ] pλ = v,, (3) where f = λ, representng the sum of request rates of all users who choose ths cloud provder. Consderng all cloud users together, Eqn. (3) s equvalent to f = f there are M cloud users. Mvµ(µ f) + Mcf (rµ c pµ)(µ f), (4) The cloud provder s problem s to maxmze ts expected proft, denoted by π. That s, max p s.t. π = pf γ(µ) (5) Mvµ(µ f) + Mcf f = (rµ c pµ)(µ f) µ > f. It s worth notng that, although both the optmal prce and the market share of the cloud provder s assumed to be unknown n ths problem, the optmzaton varable s the prce only. The reason s that the market share of the cloud provder s not an ndependent varable, and t s a functon of the prce n essence. Once the prce of the cloud provder s chosen, cloud users wll make ther decsons of whether to choose ths cloud servce or not based on ther reservaton values of the tasks. Ths mples that as long as the prce s determned, the market share s a determnstc value. Solvng the optmzaton problem (5), the optmal usage prce for the cloud provder s found to be p = max{p m, p Λ }, where Λ s referred to as the market sze and equals the sum of request rates of all cloud users n the market. In the monopoly market, Λ equals to the market share f f all users choose to µmc µr c, use the cloud provder. When the market sze Λ > µ the cloud provder s not able to take the entre market n equlbrum. The equlbrum prce p equals the frst-order prce, whch takes the form of: p m = r c µ µ Mc(rµ c) µ Mv rµ c µ rµ c Mcµ, (6) wth the correspondng market share f = µ µmc µr c. Otherwse, the cloud provder wll serve all cloud users,.e., f = Λ, and the optmal prce p s: p Λ = r c µ Mc µ(µ Λ) Mv Λ. (7) In summary, when there s only one cloud provder n the market, there exsts a unque optmal prce p = max{p m, p Λ }. The optmal market share s bounded by the cloud provder s resource capacty. If ts capacty s large enough, then the cloud provder can take the entre market. Due to the exstence of the reservaton value, the cloud provder can not ncrease the prce wthout bounds, even when t s the only provder n the market. IV. PRICE COMPETITION AMONG MULTIPLE IAAS CLOUD PROVIDERS A. The Duopoly Case As a startng pont, we frst consder the case of a duopoly cloud market, n whch two IaaS cloud provders compete wth each other, wth a smlar game theoretc analyss as the monopoly case. In ths context, we derve the relatonshp between the equlbrum prces for each cloud provder, and analyze the comparatve statcs of Nash equlbrum prces. We frst dscuss how decsons are made by cloud users n ths market. All cloud users act n a selfsh fashon so as to

5 5 maxmze ther own expected net reward. The optmal choce of cloud user s to choose the cloud provder from whch t obtans a maxmzed net reward, or to refuse to use the cloud servce f ts net reward faled to exceed ts reservaton value. That s, a cloud user wll choose a cloud provder (or the opton of choosng nether cloud provder) that acheves max{u (λ ) P, v}, =, 2. (8) ) Nash Equlbrum n a Duopoly Market: Let π be the expected proft of cloud provder. Each cloud provder seeks to maxmze π by choosng ts usage prce p, whch clearly depends on the reacton of the other cloud provder and that of all cloud users. Let π (p, p 2 ) denote the expected proft of cloud provder f t chooses a prce p gven the other cloud provder k s prce p k, k and, k =, 2. A par of prces (p, p 2) s sad to be a Nash equlbrum f t satsfes: π (p, p 2) π (p, p 2), p, π 2 (p, p 2) π 2 (p, p 2 ), p 2. In a Nash equlbrum, any cloud provder can not ncrease the expected proft by changng ts prce unlaterally. That s equvalent to say, the Nash equlbrum prce s the optmal prce a cloud provder can acheve n a market when cloud provders do not cooperate wth each other. In the equlbrum, the expected profts of both cloud provders are maxmzed, and the market s balanced dynamcally. In our subsequent analyss, we am to prove whether such equlbrum exsts n the duopoly market, and how can each cloud provder acheve the equlbrum prce f t exsts. The equlbrum prces can be found by a standard procedure of dentfyng the best response functon of each cloud provder. Let p = F (p k ) be cloud provder s optmal prce gven the usage prce p k selected by cloud provder k. A Nash equlbrum n ths duopoly competton market s then a par of prces (p, p 2 ) such that p = F (p 2 ) and p 2 = F 2 (p ),.e., an ntersectng pont of two best response functons. Take cloud provder as an example. The best response functon F can be found by assumng that cloud provder 2 s prce p 2 s gven and by solvng cloud provder s problem as follows: max p π = P γ (µ ) (9) s.t. U (λ ) P v, () U (λ ) P = U 2 (λ ) P 2, () f + f 2 = µ > f, λ Λ where P s the total payment user makes to cloud provder. Both constrants () and () come from optmzng cloud users net rewards. Constrant () ndcates that for any user to choose cloud provder, t should be offered at least the same expected net reward as ts reservaton value of the task. If ths constrant does not hold, the cloud user would prefer to fnsh ts task locally rather than usng the cloud servce. Constrant () states that n equlbrum, the expected net rewards that a cloud user can derve from dfferent cloud provders should be the same, whch prohbts any cloud user from swtchng cloud provders. Smlarly, the optmal prce of cloud provder 2 can be found by solvng ts correspondng problem, under the assumpton that the prce of cloud provder, p, s gven. max p 2 s.t. π 2 = P 2 γ 2 (µ 2 ) U 2 (λ ) P 2 v, U (λ ) P = U 2 (λ ) P 2, f + f 2 = µ 2 > f 2, λ Λ Each cloud provder wll update ts prces wth respect to the reacton of ts compettor and all cloud users, untl an equlbrum pont s reached,.e., when nether cloud provder can gan a hgher expected proft by changng ts own prce unlaterally. When cloud users are charged based on ther usage of resources, the problem of fndng cloud provder s best response functon (9) s equvalent to: max p π = p f γ (µ ) s.t. f Mvµ δ + Mcf (rµ p µ c)δ cf µ 2 δ 2 cf 2 µ δ f + f 2 = [c(µ µ 2 ) µ µ 2 (p p 2 )]δ δ 2 f + f 2 Λ µ > f, where δ = µ f. By consderng the best response problems of both cloud provders together, we derve the necessary condton for the exstence of a Nash equlbrum. Any equlbrum must satsfy the followng constrants, referred to as the frst-order necessary condton for the exstence of a Nash equlbrum, as summarzed n Lemma. Lemma : The necessary condton for a set of soluton (p, p 2, f, f 2 ) to be a Nash equlbrum s that, t should satsfy the followng constrants: Z f Y µ δ (2) f + f 2 = cq Xδ δ 2 (3) f + f 2 = Λ (4) f Qδ 2(p Z δ + Y 2 f δ 2 Mcµ f Y ) QY δ δ 2 (Y δ Mcµ ) + XZ (δ 2 + δ2 2 ) (5) f 2 Qδ (p 2 Z 2 δ 2 + Y 2 2 f 2 δ 2 2 Mcµ 2 f 2 Y 2 ) QY 2 δ δ 2 (Y 2 δ 2 Mcµ 2 ) + XZ 2 (δ 2 + δ2 2 ) (6) X(Xδ cµ µ 2 )(p r Z δ + Y 2 f δ 2 Mcµ f Y ) QY δ δ 2 (Y δ Mcµ ) + XZ (δ 2 + δ2 2 ) X(cµ µ 2 Xδ 2 )(p r 2Z 2 δ 2 + Y 2 2 f 2 δ 2 2 Mcµ 2 f 2 Y 2 ) QY 2 δ δ 2 (Y 2 δ 2 Mcµ 2 ) + XZ 2 (δ 2 + δ2 2 ), (7) (8)

6 6 where X = c(µ µ 2 ) µ µ 2 (p p 2 ), Y = rµ p µ c, Z = Mvµ 2 (µ f ) + Mvµ f, and Q = f µ 2 (µ 2 f 2 ) f 2 µ (µ f ). Proof: Eqn. (2) and (3) n Lemma are obtaned from the constrants () and () drectly by substtutng the correspondng utlty and payment functons. Constrant (2) s proved to hold wth equalty n equlbrum [5], whch gves Eqn. (4) n Lemma. Consderng the Lagrangan functon of cloud provder s problem, the requrement that the Lagrangan multplers correspondng to nequalty constrants should be greater or equal to gves Eqn. (5) and (7) n Lemma. Smlarly, by consderng the Lagrangan functon of the cloud provder 2 s problem, we have Eqn. (6) and (8). As a result, all equatons n Lemma are necessary for the exstence of a Nash equlbrum. Havng results from Lemma, we now come to the suffcent condton for the Nash equlbrum, whch s stated n Theorem. Due to the complexty of ths problem, we are not able to obtan the exact analytcal presentaton of the prce and the market share of each cloud provder n equlbrum. However, we have obtaned an mportant relaton between equlbrum prces, whch shows the prcng gap between two cloud provders. Theorem : Let (p, p 2, f, f 2 ) be a feasble soluton that satsfes all equatons n Lemma. Then (p, p 2) s a Nash equlbrum f t satsfes: p p 2 = c(µ µ 2 ) cf µ µ 2 Λµ (µ f ) + cf2 Λµ 2 (µ 2 f2 ) and where h(f ) = + h ( ˆf ), [ p k + c(µ µ k ) µ µ k c(λ f ) Λµ k (µ k Λ + f ) cf Λµ (µ f ) ] f γ (µ ), k and, k =, 2. ˆf s obtaned by solvng h (f ) =. Proof: We take the cloud provder s problem as an example to show a proof sketch. In vew of Lemma, cloud provder s problem s equvalent to max f Λ s.t. + + [ h(f ) = p 2 + c(µ µ 2 ) µ µ 2 c(λ f ) µ 2 Λ(µ 2 Λ + f ) Mv + McΛ cf f µ Λ(µ f ) ] f γ (µ ) c(λ f ) µ 2 Λ(µ 2 Λ + f ) r p 2 c, µ 2 cf µ Λ(µ f ) wth addtonal constrants on f such that f < µ and f 2 = Λ f < µ 2. By dfferentaton, we show that functon h s concave. Let ˆf be the maxmal pont of h n (Λ µ 2, µ ), ˆf s the root of h (f ) =. (9) To guarantee Inequalty (2) to be bndng n the case of two cloud provders, we have to further requre h ( ˆf ), whch completes the proof. Corollary : When µ 2 > µ, p 2 > p n equlbrum. Snce cloud users are senstve to fnshng tmes of ther tasks, a cloud provder wth a larger resource capacty s more lkely to complete a task wthn a shorter perod of tme. As a consequence, t s preferred by more cloud users. Because a larger resource capacty helps a cloud provder to enoy an advantageous poston n the competton market, the cloud provder can charge a hgher prce as a result. Proof: Gven µ 2 > µ and µ > f p 2 p = c(µ 2 µ ) cf + µ µ 2 Λµ (µ f ) cf2 Λµ 2 (µ 2 f2 ) > f µ 2 2 µ 2 f2 f f2 µ 2 + µ f2 f µ µ 2 (µ f )(µ 2 f2 ) f µ 2 (µ 2 f2 ) > µ µ 2 (µ f )(µ 2 f2 ) >. Based on the results n Theorem, we present a smple teratve algorthm that can be used to compute the prce for each cloud provder n a duopoly market, descrbed n Algorthm. Though we are unable to prove that the converged prce s guaranteed to be the Nash equlbrum, our smulaton results shown n Sec. V have shown that ths s the case n our smulaton scenaros. Algorthm Compute the prce for cloud provder n a duopoly market. : (Intalzaton). Each cloud provder sets the usage prce to be p = r v c. 2: (Iteratve step). Each cloud provder then updates ts prce usng another cloud provder s prce and the current market shares f and f : p = p + c(µ µ ) cf µ µ Λµ + cf (µ f ) Λµ (µ f ) 3: (Convergence crteron). Repeat the teratve step untl the prce p dffers from ts prevous value by less than a predetermned value ɛ. We use an example to llustrate how prces converge teratvely. Snce the operatng cost of each cloud provder does not affect ther selectons of prces, we set both γ (µ ) and γ 2 (µ 2 ) to n ths example. Suppose the reservaton value v =, the beneft factor r = 5, and the watng cost factor c =. When there are 2 cloud users n the market, the convergence of usage prces per resource unt as well as the market shares of two cloud provders wth resource capactes µ = 2 and µ 2 = 4 are shown n Fg. 2 and Fg. 3, respectvely. We set ɛ =. as the convergence crteron n ths example. As we can see, the proposed algorthm converges rapdly wthn 4 teratons. As we have shown n Corollary, the cloud provder wth a larger resource capacty charges a hgher prce. In ths case, p 2 > p. We can also observe from the

7 7 Usage Prce p 2.75 p # of Iteratons Fg. 2. prces. The convergence of usage Market Share f (%) f 2 (%) 5 # of Iteratons Fg. 3. The convergence of market shares. fgure that the cloud provder wth a larger resource capacty also attracts more cloud users n equlbrum. Ths reveals that cloud provders can become more compettve n the market by ncreasng ts resource capacty. Snce cloud users are senstve to fnshng tmes of ther tasks, even a cloud provder wth a hgher resource capacty wll charge a hgher prce, most cloud users stll prefer to choose ths faster provder. 2) Nash Equlbrum n a Duopoly Market wth Homogeneous Cloud Provders: In the homogeneous case that two cloud provders have the same resource capacty, the next theorem establshes the result that a unque Nash equlbrum exsts n the duopoly market, and that the teratve algorthm above always converges to the same prce for both cloud provders. It can be derved by solvng the optmzaton problem n Theorem. Theorem 2: When reduced to homogeneous cloud provders,.e., µ = µ 2 = µ, the Nash equlbrum (p, p 2) n Theorem takes the form of: p = p 2 = r c µ 2Mc µ(2µ Λ) 2Mv Λ wth the correspondng market shares f = f 2 = Λ/2. Proof: When the two cloud provders are equvalent to each other n servce rate,.e., µ = µ 2 = µ, the two cloud provders are ndfferent to the cloud users, whch mples that the equlbrum soluton s symmetrc [26]. Accordng to Theorem, we have p p c(f2 f ) 2 = Λ(µ f )(µ f 2 (2) ). Based on constrant (), we know that the followng relatonshp must hold between the optmal prces and market shares. f + f2 c(f f2 ) = (p p 2 )(µ f )µ f 2. (2) Substtute Eqn. (2) nto Eqn. (2), we can obtan that f + f 2 = Λ. Based on the symmetrc characterstc of the equlbrum soluton, we now have that f = f 2 = Λ/2. Now let s take cloud provder s expected proft maxmzaton problem as an example. Constrant () requres that (rµ p µ c)(µ f )f Mvµ(µ f ) + Mcf. When f = Λ/2, ths mples that p r c µ 2Mc µ(2µ Λ) 2Mv Λ. Snce the optmal prce s the one that maxmzes the expected proft of each cloud provder, whch equals p f. We get he result that the Nash equlbrum prce p = r c µ 2Mc µ(2µ Λ) 2Mv Λ. Smlarly, we can obtan that the Nash equlbrum prce p 2 = r c µ 2Mc µ(2µ Λ) 2Mv Λ as well, whch completes the proof. We can see from Theorem 2 that, when cloud provders are homogeneous,.e., they have the same resource capactes, both cloud provders wll charge the same prce, whch has the form of the monopoly prce p Λ, wth each of them takng half of the market. That s, both cloud provders behave ndependently and operate exactly the same as monopolsts. Corollary 2: The comparatve statcs of the homogeneous Nash equlbrum prce n Theorem 2 are as follows: p r p v = >, p c = 2 µc < = M p <, Λ µ = c Mc(2µ Λ) + 2µ µ µ 2 (µ Λ) 2 > ] [(µv c)λ 2 4vµ 2 Λ + 4vµ 3 p Λ = 2M µλ 2 (2µ Λ) 2 > < Λ < 2vµ2 2µ µvc < µv c 2vµ 2 2µ µvc µv c < Λ < µ µ > c v > < Λ < µ µ c v The comparatve statcs of the equlbrum prce are llustrated n Fg. 4. As we can see, they conform wth most of our ntutons. A cloud provder wll rase the usage prce wth an ncrease n the users beneft factor r and ts resource capacty µ, and reduce the prce n response to an ncrease n the watng cost factor c and the reservaton value v. Usage Prce p(x) x=r x=c x=v x=µ x= Varable x Fg. 4. An llustraton of the statcs of the homogeneous Nash equlbrum prce. Snce the beneft factor r reflects the mportance of the task and the watng cost factor c represents ts urgency, ths mples that the more mportant the task s, the more the cloud provder wll charge the user; the more urgent users vew the task, the less the cloud provder dares to charge n order to wn the deal from users. The ratonale s that f the cloud provder knows that the task s mportant to the user,.e., the user wll gan substantal beneft by completng the task, the cloud provder wll nfer that the user s wllng to pay more to fnsh

8 8 the task, and hence ask for more. Yet f the task s urgent, the cloud provder wll tend to ask less to make up for ts lmted resource capacty. In addton, f the cloud provder s able to ncrease ts resource capacty µ by nvestng n new servers or by upgradng ts current facltes, t wll be able to ask more for the mproved servce qualty. In the fgure, we can see that when the resource capacty s small, the demand wll be relatvely strong compared to the resource capacty, whch results n a non-compettve envronment n the market. Increasng the resource capacty slghtly results n a rapd ncrease of usage prce. Fnally, n cases when users have a hgher reservaton value v and thus may refuse to use the cloud servce wth a hgher probablty, cloud provders wll have to reduce ther prces to attract users. The market sze Λ wll affect the usage prce n a more complcated way. When the resource capacty s large enough,.e., µ > c v, the cloud provder wll rase the prce when the market sze ncreases, untl t reaches a certan threshold, e.g., Λ = 4.6 n the fgure. If the market sze contnues to ncrease, users wll overwhelm the cloud provders and may experence longer task fnshng tmes. Cloud provders wll reduce the prce to compensate for the ncreased watng costs. However, f the resource capacty s small,.e., µ < c v, the market s agan n a non-compettve envronment, whch results n the fact that prce ncreases wth a larger market sze. 3) Socal Welfare Problem n the Duopoly Market: We have prevously analyzed the Nash equlbrum prces n a competton market wth two cloud provders. In equlbrum, each cloud provder s prce s determned by ts best response functon to the other cloud provder s prce. In other words, prces are optmzed for cloud provders only, wth no drect mplcaton that all cloud provders and users wll reach an outcome that s socally optmal. A choce of prces one by each cloud provder s socally optmal f t maxmzes the sum of payoffs to all partcpants [27]. In the cloud market, t mples a set of equlbrum prces at whch the payoffs of both cloud provders and cloud users are maxmzed. A cloud user s payoff for beng served by cloud provder s ts expected net rewards U(λ ) P, wth a request rate of λ and a usage prce p, =, 2; a cloud provder s payoff n ths market equals ts expected proft, whch s π. Therefore, the socal welfare s: [U(λ ) P ] + π., In the socal welfare problem, prces are smply an nternal transfer of wealth and hence are not consdered as obectve varables. Our nterest s how cloud users are dstrbuted between two cloud provders to maxmze socal welfare. Though we hope that the duopoly equlbrum prces are also socally optmal, our analyss shows t s not always the case. Theorem 3: The socal welfare maxmzng soluton (f s, f 2s) s not always the same as the market shares (f, f 2 ) n equlbrum. Due to space constrants, we provde a detaled proof n our supplementary techncal report [28]. Though the concluson that the socal welfare maxmzng soluton s not the same as the market shares n equlbrum s not a surprse, t s not ntutve ether. More mportantly, a Prce of Anarchy of can be acheved n a homogeneous duopoly, whch means that the socal welfare maxmzer also reflects the equlbrum market share. B. The General Case Based on our game theoretcal analyss of prce competton n the monopoly and duopoly cloud markets, we now proceed to consder the general case when multple cloud provders are competng wth one another. Our analyses wll show that a unque Nash equlbrum exsts n an olgopoly cloud market. We wll also present an teratve algorthm to compute the equlbrum prces based on our analyses. ) Cloud Provder s Problem n an Olgopoly Market: From our prevous analyss n the duopoly market, we can see that the market share of each cloud provder s not only affected by the cloud provder s own prce, but also the other cloud provder s prcng choce, whch are both varables to be determned. In a market wth multple cloud provders, the usage prces wll nfluence each cloud provder s market share n a hghly complcated way, and due to ths reason we are not able to get the exact analytcal presentaton. As an alternatve, We apply the Multplcatve Compettve Interacton (MCI) model to capture the relatonshp between usage prces and market shares n an olgopoly market. Proposed by Bell et al. n 975 [29], the MCI model s wdely used n economc competton markets [6], [26]. To be specfc, the market share of each cloud provder n a market wth N cloud provders s assumed to take the followng form: ( L p a t b f = Λ N = L p a t b ), =, 2,..., N, where t represents the expected fnshng tme of a unt request experenced at cloud provder, ncludng both the watng tme n the queue and the servce tme. The numerator L p a t b, wth a, b, represents the attracton of cloud provder, whch corresponds to how cloud users feel towards ts servce gven ts usage prce, expected fnshng tme, and other compettve factors, e.g., API, load balancng, and reputaton. The parameter a and b are referred to as the prce attracton factor and the fnshng tme attracton factor, respectvely. The parameter L > represents the combned effects of other compettve factors, wth a larger L reflectng a hgher degree of attracton to cloud users. Cloud provder s market share s gven by ts relatve attracton to all cloud provders n the market. In subsequent analyses, we choose a = b = for smplcty. Based on queueng theory, the expected fnshng tme t of a sngle request, ncludng both the watng tme and the servce tme, equals µ f n an M/M/ queue. Note that n a cloud envronment, for a gven resource capacty µ at cloud provder, the expected fnshng tme s a functon of ts market share f, whch s determned by the cloud provder s usage prce. As a consequence, f we use β = L L p t to denote

9 9 the combned attracton of cloud provder s compettors, the market share of cloud provder s f = Λ, =, 2,..., N. β p t + Substtutng t = µ f, together wth the requrement that f < µ, we have t = 2 µ β p Λ + (µ + β p + Λ) 2 4Λµ. (22) Expressed as a functon of usage prces, the market share of cloud provder, f, s n a much more complcated form than that n typcal economc papers n the lterature dscussng prce competton, and ths makes our subsequent analyses substantally more challengng. Agan, the obectve of cloud provder s to fnd the best response functon that maxmze ts expected proft, takng nto consderaton the usage prces set by other cloud provders. Mathematcally, the problem faced by cloud provder can be formulated as the followng: max p s.t. ( p π = Λ β p t + Λ(µ r c µ p ) µ (β p t + ) Λ µ > β p t + t = ) γ (µ ) (23) cmλ µ (µ β p t + µ Λ) Mv 2 µ β p Λ + (µ + β p + Λ) 2 4Λµ. Constrants n problem (23) ensure that any cloud user who chooses cloud provder wll be offered at least the same expected net reward as ts reservaton value. 2) Nash Equlbrum n an Olgopoly Market: In an olgopoly market wth N cloud provders, the N-tuple prce vector (p, p 2,..., p N ) s called a Nash equlbrum f for each cloud provder, p s the best response to prce p chosen by all other frms. In other words, the Nash equlbrum mples that no sngle cloud provder can beneft by devatng from ths equlbrum pont unlaterally. By solvng problem (23), cloud provder s able to compute ts optmal prce p based on the combned attracton of ts compettors β, whch n turn can be computed usng the current prces p of other cloud provders. Wth the dea of solvng ths problem n an teratve fashon, we have desgned the followng teratve algorthm, Algorthm 2, to compute the Nash equlbrum prce for each cloud provder. In subsequent analyses, we are gong to show that a unque Nash equlbrum exsts n a prce competton market wth multple cloud provders, and the above teratve algorthm always converges to ths equlbrum soluton. Ths s farly sgnfcant, n that f the requred nformaton n Algorthm 2 s avalable, we now have an algorthmc tool to compute the unque Nash equlbrum. We frst present a necessary result that s useful to derve the key results n Lemma 3. Lemma 2: When µ > Λ, Mvµ 2 + cmλ > Λ2 (µ r c) and Mvµ (µ Λ) + cmλ < Λ(µ r c)(µ Λ). Algorthm 2 Compute the Nash equlbrum prce for cloud provder n an olgopoly market. : (Intalzaton). Each cloud provder sets the usage prce to be a very small value p = ɛ. 2: (Iteratve step). 3: for cloud provder to N do 4: Each cloud provder computes the optmal prce p by solvng problem (23) usng the current values p of other cloud provders, and updates the prce p. 5: end for 6: (Convergence crteron). Repeat the teratve step untl prce p dffers from ts prevous value by less than some predetermned value ɛ. Proof: We use the frst nequalty to show a proof sketch here. The second nequalty can be proved by usng the same technque. The frst nequalty n Lemma 2 s equvalent to Mvµ 2 Λ 2 rµ + Λ 2 c cmλ >. (24) It s obvous that Mv >, Λ 2 r <. Snce Λ M, Λ 2 c cmλ. If Λ 4 r 2 4Mv(Λ 2 c cmλ), then nequalty (24) holds for all µ. On the other hand, f Λ 4 r 2 4Mv(Λ 2 c cmλ) >, we fnd that Λ > µ, where µ s the larger root gven by the quadratc functon n nequalty (24). To summarze, nequalty (24) holds when µ > Λ, whch completes the proof. Based on Lemma 2, the followng Lemma establshes two key results for analyzng the prce competton n an olgopoly cloud market. To be specfc, Lemma 3 characterzes the optmal prce for each cloud provder when prces of all other provders are gven. It also reveals the fact that a cloud provder wll have to compete wth a lower prce when the combned attracton of ts compettors ncreases. Furthermore, Lemma 3 also serves as an mportant supportng lemma n the proof of Theorem 4. Lemma 3: () A unque optmal prce p exsts for cloud provder, and s bounded by < p p. () The optmal prce p s decreasng n β. Proof: Accordng to Eqn. (22), the expected fnshng tme t s a functon of the usage prce p at each cloud provder. Let H(p ) = p t. H(p ) takes the form of H(p ) = 2p µ β p Λ + (µ + β p + Λ) 2 4Λµ. Note that β s determned by the prce and combned effects of all other cloud provdes except. Once the prces of other cloud provders are fxed, β s a constant from cloud provder s pont of vew. Snce β α 4Λµ ) + 2p β α 2 4Λµ 2(α + α 2 H(p ) = p α 2 4Λµ α +, α 2 4Λµ where α = µ β p Λ, α = µ + β p + Λ, we know that H(p) p >, whch mples that H(p ) s ncreasng wth p. Fg. 5 shows how H(p ) changes wth prce p. As we can see n the fgure, f we can prove that a unquely exsted

10 H (p ) that maxmzes cloud provder s expected proft n problem (23) s bounded by [, H(p )], and s decreasng n β, we are able to prove that a unquely exsted optmal prce p for cloud provder s bounded by < p p, and s decreasng n β. H(p ) H (p ) H(p ) Fg. 5. The graph of the functon H(p ). () Based on the fact that the equlbrum market share f < µ, we have H(p ) Λ µ β µ. Snce H(p ) = p t should also be greater than to ensure a non-negatve expected proft obtaned by cloud provder, the nequalty Λ µ β µ < Λ < µ (25) must hold to avod an nfnte queueng delay. The constrants n problem (23) can be rephrased as p A H(p ) 2 + B H(p ) + C, (26) where A = Mvµ 2 β2 + µ2 Λβ, B = Mvµ β (µ Λ) + µ 2 β Mv µ β Λ(µ r c) + µ Λ(µ Λ) + cmλβ, and C = Mvµ (µ Λ) + cmλ Λ(µ r c)(µ Λ). It s obvous that A >. For a meanngful problem that s worth dscussng, each cloud provder should be able to attract at least one cloud user when t charges a usage prce and there s no other cloud users watng n the queue, whch s to say ) (r cµ Λ > Mv (rµ c)λ > Mvµ. (27) Based on Eqn. (27) and the result n Lemma 2, we have C < and B µ Λ(µ Λ),. Ths shows that the quadratc functon n (26) has two realvalue roots H(p ) and H(p ), wth H(p ) < and H(p ) >. Ths establshes the result that H (p ) s bounded by < H (p ) H(p ), whch mples that the equlbrum prce p for cloud provder s bounded. () Accordng to Eqn. (26), we have (2Mvµ 2 β + µ 2 Λ)H(p ) 2 + (Mvµ 2 β 2 + µ 2 β Λ)2H(p ) H(p ) + p [B µ Λ(µ Λ)]H(p ) + B, β β β whch mples that H(p ) β H(p ) 2 H(p) B µ Λ(µ Λ) β 2Mvµ 2 β+µ2 Λ p 2H(p ) Mvµ2 β2 +µ2 βλ 2Mvµ 2 β+µ2 Λ + B p. Ths shows that functon H(p ) s decreasng n β. Based on the fact that p H(p ) = 2(µ Λ), H(p ) [ 3 2 H(p ) β 2 2 ] 3 2 H(p ) we have p β = p H(p ) β, whch establshes the second statement that the optmal prce p s decreasng n β. We further observe that π = Λ Λp2 t β p Λ 2 p 2 β t p p (β p t + ) 2. Based on the fact that β p < and t p <, we have π p >, whch completes the statement that the optmal prce p unquely exsts. Theorem 4: Algorthm 2 always converges to the unque Nash equlbrum soluton n an olgopoly cloud market. Proof: We are now ready to show that Algorthm 2 always converges to the Nash equlbrum. Let p (k) be the optmal prce at the kth teraton. We shall prove by nducton that p (k) s ncreasng n k. Snce p s bounded by p, ths establshes the result that p (k) always converges. By defnton, p () = ɛ, where ɛ can be chosen to be small enough such that p (k) > ɛ. In partcular, when k =, p () > p (). Now assume that p (k) p (k ) for all and k < n. At the begnnng of the nth teraton, we have β (n) = L > L > L (p (n ) ) L (p (n 2) ) = β (n ), where the nequalty follows from the nductve assumpton. Then from the second statement n Lemma 3, we know that p (n) p (n ). Suppose that p (n) p (n ), for all =, 2,..., l. Then for cloud provder l at the nth teraton we have β (n) l = L l >l L l >l = β (n ) l, L (p (n) L (p (n ) As a result, we deduct that p (n) l that p (n) p (n ) ) + L l >l ) + L l L (p (n ) ) >l L (p (n 2) ) p (n ) l. Ths mples for all, whch completes our proof of convergence. We now prove that the converged Nash equlbrum must be unque by contradcton. The equlbrum result can be represented as Φ = (At, At 2,..., At N ), where At = L p t the converged equlbrum prce p represents the attracton of cloud provder gven. Snce we have proved n Lemma 3 that the optmal prce p for each cloud provder unquely exsts, the correspondng expected fnshng tme of a unt request experenced at cloud provder t s also unquely determned by Eqn. (22), whch unquely defnes each At n Φ. Let us assume that there exsts another equlbrum result that can be represented as Φ = (At, At 2,..., At N ), where each At s determned by the prce p. Express At = ( + θ )At. By numberng the cloud provders properly, we can assume that θ θ 2... θ N and θ >.

11 Our frst argument s that θ 2 >. If θ 2, t wll mply that θ, 2, whch s equvalent to say that At At, 2. Snce for cloud provder, the combned attracton of all ts compettors β = L 2 At, we have β β. Recall n our proof of Lemma 3, we proved that the functon H(p ) = p t s decreasng n β, whch mples that p t p t. We can then express the equlbrum attracton of cloud provder as At = L p t p t = At = ( + θ )At. (28) Eqn. (28) mples that θ, whch contradcts wth our assumpton that θ >. Now, let us consder the optmzaton problem of cloud provder wth the attractons of other cloud provders to be At = (+θ 2)At, 2. Let p and t be the correspondng optmal soluton solved by cloud provder, then from the frst constrant n problem (23), we have Λ(µ r c µ p ) µ (β p t + ) cmλ µ (µ β p t + µ Λ) = Mv (29) Snce Φ = (At, At 2,..., At N ) s an equlbrum result, we also have Λ(µ r c µ p ) cmλ µ (β p t + ) µ (µ β p t + µ = Mv (3) Λ) Snce we have proved that θ 2 > and At = ( + θ 2 )At, 2, we have β > β. Agan, we can represent the functons n Eqn. (29) and Eqn. (3) as A H(p ) 2 + B H(p ) + C = and A H(p ) 2 + BH(p ) + C =, where A, B, C and A, B, C take the same form as A, B, C n Eqn. (26). Snce β > β, we have A > A. Snce B β = B µλ(µ Λ) β, and we have B µ Λ(µ Λ), ths mples that B > B. Based on the fact that = Mvµ(µ Λ)+µ2 Mv µλ(µr c)+cmλ Mvµ 2 β+µ2 Λ + µλ(µ Λ) Mvµ, 2 β2 +µ2 Λβ B A we have B A Eqn. (3) that < B A Λ(µ r c µ p ) µ (β p t + ) Λ(µ r c µ p ) µ (β p t + ) whch s equvalent to. It then follows from Eqn. (29) and cmλ µ (µ β p t + µ Λ) > cmλ µ (µ β p t + µ Λ), µ β 2 (p t ) 2 + (µ Λ + )β p t > µ β 2 (p t ) 2 + (µ Λ + )βp t. (3) If we substtute β = L 2 At = (+θ2) L 2 At and β = L 2 At nto nequalty (3), we can obtan the result that µ 2 At [( + θ 2 )(p L t + p t ) + (µ Λ + )] [( + θ 2 )p t p t ] >. Ths mples that (+θ 2 )p t > p t, whch also mples that At < ( + θ 2 )At. Now let us consder the equlbrum result Φ = (At, At 2,..., At N ). Snce θ θ 2, > 2, we have At 2. Agan, from our proof of Lemma 3, we can obtan At the result that p t p t. Then we wll have the followng result that At = L p t L p t = At = ( + θ )At. Combnng the two results, t s not dffcult to have the followng result that ( + θ 2 )At > At ( + θ )At, whch means that θ 2 > θ. Ths contradcts wth our assumpton that θ θ 2, and therefore, completes our proof of the unqueness of the Nash equlbrum. Agan, when all cloud provders n the market have the same resource capacty, we are able to obtan an analytcal form of the Nash equlbrum soluton, whch can be derved by solvng the optmzaton problem (23). In the next theorem, we wsh to establsh the fact that a unque Nash equlbrum exsts n the olgopoly market of homogeneous cloud provders, and that Algorthm 2 always converges to such an equlbrum prce. Theorem 5: When reduced to homogeneous cloud provders,.e., µ = µ and L = L, for all, the Nash equlbrum wth N cloud provders (p, p 2,..., p N ) takes the form of: p = r c µ NMc µ(nµ Λ) NMv Λ, wth the correspondng market shares f = Λ/N, for all. Proof: Agan, when cloud provders are equvalent to one another, the equlbrum soluton s symmetrc [26],.e., f = f and p = p for all. From the MCI model of the market share, we know mmedately that the market share of each cloud provder equals ( ) f = Λ L p a N = L p a t b t b = Λ N = f, snce t = µ f = t for all cloud provder. Snce f also takes the form of f Λ = β +, we have β p t p t + = N. Based on the constrants n problem (23), we have Λ(µr c µp ) µn whch s equvalent to cmλ µ(µn Λ) Mv, p r c µ MvN Λ cmn µ(µn Λ), Now we can complete the proof that wth the obectve of maxmzng the expected proft, the optmal prce for each cloud provder takes the form of p = r c µ NMc µ(nµ Λ) NMv Λ, for all. By comparng wth results n Theorem 2, we can see that the Nash equlbrum n an olgopoly market s n the general form of that n a duopoly market. All cloud provders n the market wll charge the same prce that has the form of the monopoly prce p Λ, wth each of them takng /N of the market. In other words, each cloud provder wll behave ndependently and operate exactly the same as a monopolst, when all of them have the same resource capacty. The comparatve statcs of the homogeneous Nash equlbrum prce n an olgopoly market s the same as what was shown n Corollary 2.

12 2 V. EVALUATION We now present our evaluaton results based on smulatons, on how the Nash equlbrum s nfluenced by both cloud provders and cloud users. From the cloud provders perspectve, we study the effects of resource capactes on equlbrum prces. On the cloud users sde, we show how the task mportance and urgency can nfluence the equlbrum prces of the cloud servce. The desgn of our smulator s based on a tme-slotted synchronous model, wth all events generated and processed n ther respectve tme slots. Our smulator s developed n the MATLAB envronment. Usage Prce p 2 f 2 (%) µ 2 Market Share µ 2 /µ Fg. 6. Effects of resource capactes Fg. 7. Effects of resource capactes on a cloud user s equlbrum prce on the rato of equlbrum prces and and ts market share. market shares. Rato of Equlbrum Prces p 2 /p f 2 /f Rato of Market Shares A. Analyzng the Nash Equlbrum n a Duopoly Market We begn our evaluaton wth two cloud provders competng n the market. Snce the proposed algorthm s shown to be able to fnd the Nash equlbrum prces wthn a small number of teratons, the equlbrum prces n each smulated scenaro are obtaned by Algorthm. Our smulaton results have further valdated our analytcal results n Sec. IV. We assume that there are 2 cloud users n the market,.e., M = 2. Except otherwse specfed, the resource capacty of each cloud provder s set to be µ = 2 and µ 2 = 4; the operatng costs γ (µ ) and γ 2 (µ 2 ) are set to be to focus only on the prce competton; the reservaton value v s set to be ; the beneft factor r s set to be 5; and the watng cost factor c s set to be for all cloud users. User s request rate λ for the cloud servce s unformly random n (,.5), as the total request rate has to be smaller than the total servce rate to avod an unlmted queueng delay. Effects of resource capactes on equlbrum prces: We frst study how a cloud provder s resource capacty, µ, wll affect the Nash equlbrum prces. In ths scenaro, µ s set to be 2, µ 2 > µ and ther rato s assumed to range from to 4. Fg. 6 shows how the Nash equlbrum prce and the market share of cloud provder 2 react when ts server capacty ncreases, whle that of cloud provder remans the same. As we can see, both the usage prce and the market share ncrease wth the resource capacty. To further understand the mpact of resource capactes on both cloud provders, we compute the rato of Nash equlbrum prces as well as the rato of market shares of the two cloud provders. Our results n Fg. 7 show that, when resource capactes change whle other characterstcs reman constant, the comparatve advantage of cloud provder 2 to cloud provder on both prce and the market share also ncreases, whch further proves the mportance of the resource capacty. Effects of resource capactes on expected profts: Snce the obectve of cloud provders s to maxmze ther expected profts, we now dscuss how expected profts are nfluenced by resource capactes. In our dscusson, the expected proft of a cloud provder equals ts usage prce tmes ts market share. Shown n Fg. 8, the expected proft of cloud provder 2 does not ncrease evdently after a certan threshold, and may even decrease, wth lnear, exponental and step functons for operatng costs. The ratonale s that when the resource capacty ncreases, the cloud provder s able to charge a hgher prce and attract more users, and hence s able to generate more proft. However, f the cloud provder contnues to ncrease ts resource capacty, the operatng cost, such as power consumpton, may be too large to be made up by the ncreased revenue, whch may result n reduced proft. Expected Proft (µ) = a + bµ (µ) = a k µ (µ) = a u(µ ) µ 2 Fg. 8. Effects of resource capactes on a cloud provder s expected proft wth dfferent operatng cost functons (a =, b = 8, and k =.4). B. Analyzng the Nash Equlbrum n an Olgopoly Market We are now ready to use our algorthm n fndng the Nash equlbrum prces to derve some nterestng observatons, as multple cloud provders are competng for the same pool of cloud users. Our evaluaton s conducted n a scenaro wth 4 cloud provders, wth resource capactes of µ = 5, µ 2 = 25, µ 3 = 2, and µ 4 = 5, respectvely. To focus on the prce competton only, the operatng costs are set to be zero, and the combned effects of other factors are set to be L = L 2 = L 3 = L 4 =. Wth the same number of 2 cloud users n the market, the reservaton value v s set to be ; the beneft factor r s set to be 2; and the watng cost factor c s set to be for all cloud users. User s request rate λ for the cloud servce s unformly random n (, ). Usage Prce # of Iteratons p p 2 p 3 p 4 Fg. 9. Fndng the equlbrum prces n a cloud market. Market Share C C2 C3 C4 Cloud Provders 5 5 Expected Revenue Fg.. The Nash equlbrum market shares and expected revenues of each cloud provder.

13 3 Fg. 9 shows the prce of each cloud provder at each teraton. As we can see, when we set the ntal prce and the convergence crtera of four cloud provders to be., our proposed algorthm can fnd the Nash equlbrum prces n ths market wthn 7 teratons. The Nash equlbrum has further confrmed our concluson that the cloud provder wth a larger resource capacty s able to charge a hgher prce n a competton market. In addton, Fg. shows the average market shares and expected revenues of each cloud provder, together wth the correspondng 95% confdence nterval. It s nterestng to observe that, though equpped wth very dfferent resource capactes (e.g., C4 offers twce the capacty over C2), each of the four cloud provders takes approxmately an equal amount of the market share. Ths shows that doublng the resource capacty may not necessarly ncrease the market share by any sgnfcant margn. Wth respect to the expected revenue, a larger resource capacty does lead to more revenue; but ths may not drectly translate to hgher profts, due to the escalatng captal and operaton expenses assocated wth larger capactes. We wll dscuss more mplcatons of these observatons n the next secton. # of Iteratons Intalzed Prce ( x ) Fg.. Effects of the ntal prce on equlbrum prces. Relatve Dfference 3 2 Prce Market share Proft Rato of Other Factors Fg. 2. Effects of the combned attracton of other factors. Snce the ntal prce of each cloud provder may affect the number of teratons that our algorthm needs to acheve the Nash equlbrum, we are nterested n how ths ntal prce wll affect the convergence of the proposed algorthm. Shown n Fg., when the ntal prce vares from to 5, the number of teratons our algorthm requres to converge has been very mldly affected, only ncreasng from 5 to 8. As a result, even f we set the ntal prce of each cloud provder to be a very small number to avod mssng the Nash equlbrum prce, our algorthm can always converge wthn a few teratons. As ntroduced n Sec. IV-B, the Multplcatve Compettve Interacton (MCI) model s used n our analyss to capture the relatonshp between usage prces and market shares n an olgopoly market. It has ncorporated the ablty to represent how cloud users feel towards the servce of a cloud provder, ncludng alternatve nfluental factors such as whether the Applcaton Programmng Interface (API) s secure and easy to use, how load balancng s to be performed, as well as the reputaton and brand of the cloud provder. As the fnal experment n ths secton, we are nterested n nvestgatng the combned effects of these alternatve factors. We use two cloud provders as an example, wth resource capactes µ = 5 and µ 2 = 5. Fg. 2 shows when the rato of L2 L vares from. to, how relatve dfferences of ther usage prces, ther market shares, and ther expected profts change accordngly. As the rato becomes smaller, t represents the fact that the provder wth a larger capacty has become less attractve to cloud users due to the combned effects of the alternatve factors. As we can observe from the fgure, beng less attractve to cloud users does not affect the usage prces, as the two cloud provders have the same prces over the entre range of ratos (.e., the relatve dfference remans zero). That sad, as the provder wth a larger capacty has become less attractve due to alternatve factors, both ts market share and ts expected proft decrease sgnfcantly, to the pont that they can be smaller than ts compettor wth a thrd of the capacty. Ths mples that f a cloud provder wshes to keep ts compettve edge, t needs to become more attractve wth respect to alternatve qualty factors, such as ts reputaton and brand. We wll dscuss more mplcatons n our concludng remarks. VI. DISCUSSIONS AND CONCLUDING REMARKS In ths paper, we have studed the problem of prce competton n a market wth multple IaaS cloud provders. In partcular, we have focused on answerng the queston: when multple IaaS provders face a common pool of potental users, how should each one of them choose the optmal prce that maxmzes ts own proft? Intutvely, f prces are set to be too hgh, cloud users wll choose alternatve cloud provders; but f they are too low, the overwhelmng demand for resources from a large number of cloud users may ncrease the task fnshng tmes, therefore negatvely affectng the performance of cloud applcatons and the utlty of cloud users. By modelng each provder as an M/M/ queue, we analyze ths problem usng a game theoretc technque n monopoly, duopoly and olgopoly markets. We have derved the suffcent condton for the exstence of a Nash equlbrum and propose two teratve algorthms for each provder to fnd ts equlbrum prce n the duopoly and olgopoly market, respectvely. Our algorthms represent a frst step towards desgnng practcal mechansms to prce resources n operatonal IaaS cloud provders, and are shown to converge quckly. One mportant queston that s closely related to our analyses and evaluaton remans: What an IaaS cloud provder, ether an establshed one or a new player makng ts market debut, should do to attract new customers and to stay compettve? By analyzng the Nash equlbrum n an olgopoly market where multple IaaS provders compete, our evaluaton have ponted to some ntrgung observatons that are worth dscussng. At a frst glance on our evaluaton results, to become more compettve n the market and to gan a larger market share, a cloud provder may ntally choose to ncrease ts resource capacty. Yet, the Total Cost of Ownershp (TCO), ncludng captal expenses (CAPEX) and operatng expenses (OPEX), escalates as cloud provders add to ther resource capactes. Such escalatng costs may become an mportant contrbutng factor that leads to much smaller margnal gans, or even margnal losses, n profts at the IaaS cloud provders. Our results n Sec. V-B have provded some nterestng nsghts for further dscussons on mprovng the proft of

14 4 an IaaS cloud provder. As shown n Fg., a larger IaaS provder wth twce the resource capacty may only gan a very margnal ncrease n ts market share. Coupled wth the hgher TCO n mantanng hgher capactes, a cloud provder should defntely consder to mprove other alternatve dmensons of qualty n ts cloud servces, ncludng securty, prvacy polces, avalablty of servces, data consstency, and ultmately, better servce-level agreements that guarantee the performance n a combnaton of servce-level obectves. The aggregate of addtonal dmensons n ts qualty of servce wll be reflected over tme n the reputaton or brand value of a cloud provder, whch wll lead to a hgher market share and proft. In Sec. V-B and Fg. 2 n partcular, thanks to the fact that our MCI model s able to capture the effects of alternatve factors, we have shown that the market share and proft of a much larger cloud provder wth more than trple the resource capacty may be smlar to ts much smaller compettor, f the combned effect of the other alternatve qualty dmensons s fve tmes worse. Ths observaton provdes a strong motvaton for an IaaS cloud provder to offer servce-level agreements wth regards to a number of servce-level obectves (e.g., avalablty and response tmes) that are weghted more heavly n the cloud user s favour, and to charge hgher prces accordngly. Whle queueng theoretc models are not sutable to provde explct gudelnes on how stronger guarantees may be provded by the IaaS provder on securty, prvacy, avalablty, and data consstency, they are nevertheless mportant factors that contrbute to the brand and reputaton of the provder. REFERENCES [] Amazon EC2 Prcng. [2] GoGrd VM Prcng. [3] RackSpace VM Prcng. 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Yuan Feng receved her B.Engr. from the School of Telecommuncatons, Xdan Unversty, X an, Chna, n 28, and both her M.A.Sc. and Ph.D. degrees from the Department of Electrcal and Computer Engneerng, Unversty of Toronto, Canada, n 2, and 23, respectvely. She s currently an Assstant Professor at the Department of Computng, Hong Kong Polytechnc Unversty. Her research nterests nclude optmzaton and desgn of largescale dstrbuted systems and cloud servces. Baochun L s a Professor at the Department of Electrcal and Computer Engneerng at the Unversty of Toronto, and holds the Bell Unversty Laboratores Endowed Char n Computer Engneerng. Hs research nterests nclude large-scale multmeda systems, cloud computng, peer-to-peer networks, applcatons of network codng, and wreless networks. He s a member of ACM and a senor member of IEEE. Bo L s a Professor n the Department of Computer Scence and Engneerng, Hong Kong Unversty of Scence and Technology. He holds a Chang Jang Vstng Char Professorshp n Shangha Jao Tong Unversty (2-23), and s the Chef Techncal Advsor for ChnaCache Corp., the largest CDN operator n Chna. Hs current research nterests nclude content dstrbuton, datacenter networkng, cloud computng, moble applcatons, and vehcular networks. He receved four best paper awards from IEEE, and s a Fellow of the IEEE.