CLoud computing has recently emerged as a new

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1 1 A Framework of Prce Bddng Confguraons for Resource Usage n Cloud Compung Kenl L, Member, IEEE, Chubo Lu, Keqn L, Fellow, IEEE, and Alber Y. Zomaya, Fellow, IEEE Absrac In hs paper, we focus on prce bddng sraeges of mulple users compeon for resource usage n cloud compung. We consder he problem from a game heorec perspecve and formulae no a non-cooperave game among he mulple cloud users, n whch each cloud user s nformed wh ncomplee nformaon of oher users. For each user, we desgn a uly funcon whch combnes he ne prof wh me effcency and ry o maxmze s value. We desgn a mechansm for he mulple users o evaluae her ules and decde wheher o use he cloud servce. Furhermore, we propose a framework for each cloud user o compue an approprae bddng prce. A he begnnng, by relaxng he condon ha he allocaed number of servers can be fraconal, we prove he exsence of Nash equlbrum soluon se for he formulaed game. Then, we propose an erave algorhm IA, whch s desgned o compue a Nash equlbrum soluon. The convergency of he proposed algorhm s also analyzed and we fnd ha converges o a Nash equlbrum f several condons are sasfed. Fnally, we revse he obaned soluon and propose a near-equlbrum prce bddng algorhm N PBA o characerze he whole process of our proposed framework. The expermenal resuls show ha he obaned near-equlbrum soluon s close o he equlbrum one. Index Terms Cloud compung, Nash equlbrum, Non-cooperave game heory, Prce bddng sraegy. 1 INTRODUCTION 1.1 Movaon CLoud compung has recenly emerged as a new paradgm for a cloud provder o hos and delver compung resources or servces o enerprses and consumers [1]. Usually, he provded servces manly refer o Sofware as a Servce SaaS, Plaform as a Servce PaaS, and Infrasrucure as a Servce IaaS, whch are all made avalable o he general publc n a pay-as-yougo manner [2], [3]. In mos sysems, he servce provder provdes he archecure for mulple users o bd for resource usage [4], [5]. When makng bds for resource usage n cloud, mulple users and he cloud provder need o reach an agreemen on he servce level and he coss o use he provded resources durng he reserved me slos, whch could lead o a compeon for he usage of lmed resources [6]. Therefore, s mporan for a user o confgure an approprae bddng prce for resource usage durng hs/her reserved me slos whou complee nformaon of hose oher users, such ha hs/her uly s maxmzed. For a cloud provder, he ncome.e., he revenue s he charge from users for resource usage [7], [8]. When provdng compung resources o mulple cloud users, Kenl L, Chubo Lu, and Keqn L are wh he College of Informaon Scence and Engneerng, Hunan Unversy, and Naonal Supercompung Cener n Changsha, Hunan, Chna, E-mal: lkl@hnu.edu.cn, luchubo@hnu.edu.cn, lk@newpalz.edu. Keqn L s also wh he Deparmen of Compuer Scence, Sae Unversy of New York, New Palz, New York 12561, USA. A. Y. Zomaya s wh he School of Informaon Technologes, U- nversy of Sydney, Sydney, NSW 2006, Ausrala. E-mal: alber.zomaya@sydney.edu.au. a suable resource allocaon model referrng o bddng prces should be sgnfcanly aken no accoun. The reason les n ha an approprae resource allocaon model referrng o bddng prces s no jus for he prof of a cloud provder, bu for he appeals o more cloud users n he marke o use cloud servce. Specfcally, f he per resource usage bddng prce s oo hgh, even hough he allocaed compung resource s enough, a user may refuse o use he cloud servce due o he hgh paymen, and choose anoher cloud provder or jus fnsh hs/her requess locally. On he oher hand, f he per resource usage charge s low whle he allocaed compung resource s no suffcenly enough, hs wll lead o poor servce qualy long ask response me and hus dssasfes he cloud users even for poenal users n he marke. Hence, a cloud provder should desgn an approprae resource allocaon model consderng users bddng prces. A raonal user wll choose a bddng sraegy o use resources ha maxmzes hs/her own ne reward,.e., he uly obaned by choosng he cloud servce mnus he paymen [1]. On he oher hand, he uly of a user s no only deermned by he mporance of hs/her asks.e., how much benef he user can receve by fnshng he asks, bu also closely relaed o he urgency of he ask.e., how quckly can be fnshed. The same ask, such as runnng an onlne voce recognon algorhm, s able o generae more uly for a cloud user f can be compleed whn a shorer perod of me n he cloud [1]. However, consderng he energy savng and economc reasons, s rraonal for a cloud provder o provde enough compung resources o sasfy all requess n a me slo. Therefore, mulple cloud users have o compee for resource usage. Snce he bddng

2 2 prce and allocaed compung resources of each user are affeced by hose decsons of oher users, s naural o analyze he behavor of such sysems as a sraegc games [9]. 1.2 Our Conrbuons In hs paper, we focus on prce bddng sraeges of mulple users compeon for resource usage n cloud compung. We consder he problem from a game heorec perspecve and formulae no a non-cooperave game among he mulple cloud users, n whch each cloud user s nformed wh ncomplee nformaon of oher users. For each user, we desgn a uly funcon whch combnes he ne prof wh me effcency and ry o maxmze s value. We sudy he conflcs of he mulple users wh neracve decsons and propose a near-equlbrum prce bddng algorhm N PBA o confgure approprae bddng sraegy for each of he users. We also perform exensve expermens o verfy he effecveness of our proposed prce bddng algorhm. In summary, he man conrbuons of hs work can be lsed as follows: We propose a framework for each cloud user o confgure an approprae bddng prce for resource usage n cloud compung. By relaxng he condon ha he allocaed number of servers can be fraconal, we prove he exsence of Nash equlbrum soluon se for he formulaed game and propose an erave algorhm IA o compue a Nash equlbrum soluon. The convergency of he proposed IA algorhm s analyzed and we fnd ha converges o a Nash equlbrum f several condons are sasfed. We revse he obaned soluon and propose a near-equlbrum prce bddng algorhm N PBA o characerze he whole process of our proposed framework. The expermenal resuls show ha he obaned nearequlbrum soluon s close o he equlbrum one, whch valdaes he effecveness of our proposed N PBA algorhm. The res of he paper s organzed as follows. In Secon 2, we presened he relevan works. Secon 3 descrbes he models of he sysem and presens he problem o be solved. Secon 4 formulaes he problem no a noncooperave game and propose a near-equlbrum prce bddng algorhm. Many analyses are also presened n hs secon. Secon 5 s developed o verfy our heorecal analyss and show he effecveness of our proposed algorhm. We conclude he paper wh fuure work n Secon 6. 2 RELATED WORKS In many scenaros, a servce provder provdes he archecure for users o bd for resource usage [5], [10], [11]. One of he mos mporan aspecs ha should be aken no accoun by he provder s s resource allocaon model referrng users bddng prces, whch s closely relaed o s prof and he appeals o marke users. Many works have been done on resource allocaon scheme referrng o bddng prces n he leraure [5], [10], [11], [12], [13], [14]. In [10], Samm e al. focused on resource allocaon n cloud ha consders he benefs for boh he users and provders. To address he problem, hey proposed a new resource allocaon model called combnaoral double aucon resource allocaon CDARA, whch allocaes he resources accordng o bddng prces. In [5], Zaman and Grosu argued ha combnaoral aucon-based resource allocaon mechansms are especally effcen over he fxed-prce mechansms. They formulaed resource allocaon problem n clouds as a combnaoral aucon problem and proposed wo solvng mechansms, whch are exensons of wo exsng combnaoral aucon mechansms. In [11], he auhors also presened a resource allocaon model usng combnaoral aucon mechansms. Smlar sudes and models can be found n [12], [13], [14], [15]. However, all of hese models only ry o mprove he profs of he cloud provders or cloud users. They faled o confgure opmal bddng prces for mulple users or show how her bddng sraeges closer o he opmal ones. Game heory s a feld of appled mahemacs ha descrbes and analyzes scenaros wh neracve decsons [16], [17], [18]. I s a formal sudy of conflcs and cooperaon among mulple compeve users [19] and a powerful ool for he desgn and conrol of mulagen sysems [20]. There has been growng neres n adopng cooperave and non-cooperave game heorec approaches o modelng many problems [21], [22], [23], [26]. In [26], Mohsenan-Rad e al. used game heory o solve an energy consumpon schedulng problem. In her work, hey proved he exsence of he unque Nash equlbrum soluon and hen proposed an algorhm o oban. They also analyzed he convergence of her proposed algorhm. Even hough he formas for usng game heory n our work,.e.rovng Nash equlbrum soluon exsenceroposng an algorhm, and analyzng he convergence of he proposed algorhm, are smlar o [26], he formulaed problem and he analyss process are enrely dfferen. In [24], he auhors used cooperave and non-cooperave game heory o analyze load balancng for dsrbued sysems. Dfferen from her proposed non-cooperave algorhm, we solve our problem n a dsrbued erave way. In our prevous work [25], we used non-cooperave game heory o address he schedulng for smple lnear deerorang jobs. For more works on game heory, he reader s referred o [24], [27], [28], [29], [30]. 3 SYSTEM MODEL AND PROBLEM FORMULA- TION To begn wh, we presen our sysem model n he conex of a servce cloud provder wh mulple cloud

3 3 users, and esablsh some mporan resuls. In hs paper, we are concerned wh a marke wh a servce cloud provder andncloud users, who are compeng for usng he compung resources provded by he cloud provder. We denoe he se of users as N {1,...,n}. Each cloud user wans o bd for usng some servers for several fuure me slos. The arrval requess from cloud user N s assumed o follow a Posson process. The cloud provder consss of mulple zones. In each zone, here are many homogeneous servers. In hs paper, we focus on he prce bddng for resource usage n a same zone and assume ha he number of homogeneous servers n he zone s m. The cloud provder res o allocae cloud user N wh m servers whou volang he consran N m m. The allocaed m servers for cloud user N are modeled by an M/M/m queue, only servng he requess from user for fuure me slos. We summarze all he noaons used n hs secon n he noaon able see Secon 1 of he supplemenary maeral. 3.1 Bddng Sraegy Model As menoned above, he n cloud users compee for usng he m servers by bddng dfferen sraeges. Specfcally, each cloud user responds by bddng wh a per server usage prce p.e., he paymen o use one server n a me slo and he number of me slos o use cloud servce. Hence, he bd of cloud user N s an ordered par b p,. We assume ha cloud user N bds a prce p P, where P [ p, p ], wh p denong user s maxmal possble bddng prce. p s a conservave bddng prce, whch s deermned by he cloud provder. If p s greaer han p, hen P s empy and he cloud user N refuses o use cloud servce. As menoned above, each cloud user N bds for usng some servers for fuure me slos. In our work, we assume ha he reserved me slos s a consan once deermned by he cloud user. We defne user s N reques profle over he fuure me slos as follow: λ λ 1 T,,...,λ 1 where λ T wh T {1,..., }, s he arrval rae of requess from cloud user n he -h me slo. The arrval of he requess n dfferen me slos of are assumed o follow a Posson process. 3.2 Server Allocaon Model We consder a server allocaon model movaed by [31], [32], where he allocaed number of servers s proporonal farness. Tha s o say, he allocaed share of servers s he rao beween he cloud user s produc value of hs/her bddng prce wh reserved me slos and he summaon of all produc values from all cloud users. Then, each cloud user N s allocaed a poron of servers as p m b,b j N p m, 2 j j whereb b 1,...,b 1,b +1,...,b n denoes he vecor of all users bddng profle excep ha of user, and x denoes he greaes neger less han or equal o x. We desgn a server allocaon model as Eq. 2 for wo consderaons. On one hand, f he reserved me slos o use cloud servce s large, he cloud provder can charge less for one server n a un of me o appeal more cloud users,.e., he bddng prcep can be smaller. In addon, for he cloud user N, he/she may be allocaed more servers, whch can mprove hs/her servce me uly. On he oher hand, f he bddng prce p s large, hs means ha he cloud user N wans o pay more for per server usage n a un of me o allocae more servers, whch can also mprove hs/her servce me uly. Ths s also benefcal o he cloud provder due o he hgher charge for each server. Therefore, we desgn a server allocaon model as Eq. 2, whch s proporonal o he produc of p and. 3.3 Cloud Servce Model As menoned n he begnnng, he allocaed m servers for cloud user N are modeled as an M/M/m queue, only servng he requess from cloud user for fuure me slos. The processng capacy of each server for requess from cloud user N s presened by s servce rae µ. The requess from cloud user N n -h T me slo are assumed o follow a Posson process wh average arrval rae λ. Le πk be he probably ha here are k servce requess wang or beng processed n he -h me slo and ρ λ /m µ be he correspondng servce u- lzaon n he M/M/m queung sysem. Wh reference o [7], we oban { 1 πk k! m ρ k π0, k < m ; m m ρ k m! π0, k m 3 ; where π 0 { m 1 l0 } 1 1 m ρ l 1 l! + m! m ρ m 1 ρ. 4 The average number of servce requess n wang or n execuon n -h me slo s N k0 kπ k π m 1 ρ m ρ + ρ 1 ρ Π, 5 where Π represens he probably ha he ncomng requess from cloud user N need o wa n queue n he -h me slo.

4 4 Applyng Lle s resul, we ge he average response me n he -h me slo as T N λ 1 λ m ρ + ρ 1 ρ Π. 6 In hs work, we assume ha he allocaed servers for each cloud user wll lkely keep busy, because f no so, a user can bd lower prce o oban less servers such ha he compung resources can be fully ulzed. For analycal racably, Π s assumed o be 1. Therefore, we have T N λ 1 λ m ρ + ρ 1 ρ µ m µ λ. 7 Noe ha he reques arrval rae from a user should never exceed he oal processng capacy of he allocaed servers. In our work, we assume ha he remanng processng capacy for servng user N s a leas σµ, where σ s a relave small posve consan. Tha s, f λ > m σµ, cloud user N should reduce hs/her reques arrval rae o m σµ. Oherwse, server crash would be occurred. Hence, we have T 1 µ + 1 m µ χ, 8 where χ s he mnmum value of λ and m σµ,.e., χ mn{λ,m σµ }. 3.4 Archecure Model In hs subsecon, we model he archecure of our proposed framework o prce bds for resource usage n cloud compung. The mulple users can make approprae bddng decsons hrough he nformaon exchange module. As shown n Fg. 1, each cloud user N s equpped wh a uly funcon u, he reques arrval rae over reserved me slos λ, and he bddng confguraon b,.e., he paymen sraegy for one server n a un of me and he reserved me slos. Le Ξ N be he aggregaed paymen from all cloud users for usng a server, hen we have Ξ N n p. Denoe 1 m m N as he server allocaon vecor, b b N as he correspondng bds, and u u N as he uly funcons of all cloud users. The cloud provder consss of m homogeneous servers and communcaes some nformaon e.g., conservave bddng prce p, curren aggregaed paymen from all cloud users for usng a server Ξ N wh mulple users hrough he nformaon exchange module. When mulple users ry o make prce bddng sraeges for resource usage n he cloud provder, hey frs ge nformaon from he nformaon exchange module, hen confgure proper bddng sraeges b such ha her own ules u are maxmzed. Afer hs, hey send he updaed sraeges o he cloud provder. The procedure s ermnaed when he se of remanng cloud users, who prefer o use he cloud servce, and her correspondng bddng sraeges are kep fxed. 3.5 Problem Formulaon Now, le us consder user s N uly n me slo T. A raonal cloud user wll seek a bddng sraegy o maxmze hs/her expeced ne reward by fnshng he requess,.e., he benef obaned by choosng he cloud servce mnus hs/her paymen. Snce all cloud users are charged based on her bddng prces and allocaed number of servers, we denoe he cloud user s paymen n me slo by P b,b, where P b,b p m b,b wh b b 1,...,b 1,b +1,...,b n denong he vecor of all users bddng profle excep ha of user. Denoe P T b,b as he aggregaed paymen from all cloud users,.e., he revenue of he cloud provder. Then, we have P T b,b n 1 1 P b,b n p m b,b. 9 On he oher hand, snce a user wll be more sasfed wh much faser servce, we also ake he average response me no accoun. From Eq. 8, we know ha he average response me of user N s mpaced by m and χ, where χ mn{λ,m σµ }. The former s vared by b,b, and he laer s deermned by λ and m. Hence, we denoe he average response 1 me of user as T b,b,λ of user N n me slo s defned as u. More formally, he uly b,b,λ r χ δ P b,b w T b,b,λ, 10 where χ s he mnmum value of λ and m b,b σµ,.e., χ mn{λ,m b,b σµ } wh σ denong a relave small posve consan, r r > 0 s he benef facor he reward obaned by fnshng one ask reques of user, δ δ > 0 s he paymen cos facor, and w w > 0 s he wang cos facor, whch reflecs s urgency. If a user N s more concerned wh servce me uly, hen he assocaed wang facor w mgh be larger. Oherwse, w mgh be smaller, whch mples ha he user s more concerned wh prof. Snce he reserved server usage me s a consan and known o cloud user N, we use u p,b,λ nsead of u b,b,λ. For furher smplcy, we use P and T o denoe P b,b and T b,b,λ, respecvely. Followng he adoped bddng model, he oal uly obaned by user N over all me slos can hus be gven by u p,b,λ 1 1 u p,b,λ r χ P w T. 11

5 5 Cloud Users Cloud Provder Confguraon: 1 1, b1, u1 Confguraon: 2 2, b2, u2 Informaon Exchange Module m 1 Confguraon: 3 3, b3, u3 Compung Requess Allocaor for reques λ 1 λ 2 λ n m 2 Confguraon: 4 4, b4, u4 Confguraon: n n, bn, un m n Fg. 1: Archecure model In our work, we assume ha each user N has a reservaon value v. Tha s o say, cloud user wll prefer o use he cloud servce f u p,b,λ v and refuse o use he cloud servce oherwse. We consder he scenaro where all users are selfsh. Specfcally, each cloud user res o maxmze hs/her oal uly over he fuure me slos,.e., each cloud user N res o fnd a soluon o he followng opmzaon problem OPT : maxmze u p,b,λ P. 12 Remark 3.1. In fndng he soluon o OPT, he bddng sraeges of all oher users are kep fxed. In addon, he number of reserved me slos once deermned by a user s consan. So he varable n OPT s he bddng prce of cloud user,.e.. 4 GAME FORMULATION AND ANALYSES In hs secon, we formulaed he consdered scenaro no a non-cooperave game among he mulple cloud users. By relaxng he condon ha he allocaed number of servers for each user can be fraconal, we analyze he exsence of a Nash equlbrum soluon se for he formulaed game. We also propose an erave algorhm o compue a Nash equlbrum and hen analyze s convergence. Fnally, we revse he obaned Nash equlbrum soluon and propose an algorhm o characerze he whole process of he framework. 4.1 Game Formulaon Game heory sudes he problems n whch players ry o maxmze her ules or mnmze her dsules. As descrbed n [5], a non-cooperave game consss of a se of players, a se of sraeges, and preferences over he se of sraeges. In hs paper, each cloud user s regarded as a player,.e., he se of players s he n cloud users. The sraegy se of player N s he prce bddng se of user,.e., P. Then he jon sraegy se of all players s gven by P P 1 P n. As menoned before, all users are consdered o be selfsh and each user N res o maxmze hs/her own uly or mnmze hs/her dsuly whle gnorng hose of he ohers. Denoe p,b,λ δ P +w T r χ. 13 ψ In vew of 11, we can observe ha user s opmzaon problem OPT s equvalen o mnmze f p,b,λ 1 ψ p,b,λ, s.. p,p P. 14 The above formulaed game can be formally defned by he uple G P,f, where f f 1,...,f n. The am of cloud user N, gven he oher players bddng sraeges b, s o choose a bddng prce p P such ha hs/her dsuly funcon f p,b,λ s mnmzed. Defnon 4.1. Nash equlbrum. A Nash equlbrum of he formulaed game G P,f defned above s a prce bddng profle p such ha for every player N : p argmn p P f p,b,λ P. 15 A he Nash equlbrum, each player canno furher decrease s dsuly by choosng a dfferen prce bddng sraegy whle he sraeges of oher players are fxed. The equlbrum sraegy profle can be found when each player s sraegy s he bes response o he sraeges of oher players. 4.2 Nash Equlbrum Exsence Analyss In hs subsecon, we analyze he exsence of Nash equlbrum for he formulaed game G P,f by relaxng one condon ha he allocaed number of servers

6 6 for each user can be fraconal. Before addressng he equlbrum exsence analyss, we show wo properes presened n Theorem 4.1 and Theorem 4.2, whch are helpful o prove he exsence of Nash equlbrum for he formulaed game. Theorem 4.1. Gven a fxed b and assumng ha r w / σ 2 µ 2 N, hen each of he funconsψ p,b,λ T s convex n p P. Proof: Obvously, ψ p,b,λ T s a real connuous funcon defned on P. The proof of hs heorem follows f we can show ha p 1,p 2 P, θp1 +1 θp 2,b,λ ψ θψ p1,b,λ +1 θψ p2,b,λ, where 0 < θ < 1. Noce ha, ψ p,b,λ s a pecewse funcon and he breakpon sasfes m σµ λ. Then, we oban he breakpon as p m Ξ ΞN\{} λ +σµ Ξ N\{} m m m σµ λ, where Ξ N\{} denoes he aggregaed paymen from all cloud users n N excep of user,.e., Ξ N\{} j N,j p. Nex, we dscuss he convexy of he funcon ψ p,b,λ. Snce p,b,λ δ P +w T r χ, ψ where χ mn{m σµ,λ }, we have ψ p,b,λ P δ +w T χ r. p p p p On he oher hand, snce T p 0 for p [ p,p and χ p 0 for p p, p ], we oban { P ϕ p,b,λ p δ χ p r p < p ; P δ p +w T p > p. Namely, ϕ p,b,λ p mp δ Ξ N\{} +m Ξ 2 N mp δ Ξ N\{} +m where Ξ 2 N We can furher oban p,b,λ 2 p 2 ψ mrµξ N\{} Ξ 2 < p ; N mwµξ N\{} m µ λ 2 > p Ξ 2, N Ξ N Ξ N\{} +p j N p j j. 2m Ξ N\{} rµ p Ξ 2 N 2m Ξ N\{} 1 p Ξ 2 N 2mw µ 2 Ξ N\{} m µ λ 2 Ξ 3 N Ξ N +1 < p ; Ξ N + µ Ξ N\{} m µ λ Ξ N +1 > p. Obvously, 2 p 2 ψ p,b,λ > 0, for all p [ p,p and p p, p ]. Therefore, p 1,p 2 [ p,p or p1,p 2 p, p ], ψ θp1 +1 θp 2,b,λ θψ p1,b,λ +1 θψ p2,b,λ, where 0 < θ < 1. Nex, we focus on he suaon where p 1 [ p,p and p 2 p, p ]. Snce ψ p,b,λ s convex on [ p,p and p, p ], respecvely. We only need o prove ha he value of ψ p,b,λ s less han ha of n he lnear funcon value conneced by he pon n p 1 and he pon n p 2,.e., p,b,λ ψ θ ψ p1,b,λ + 1 θ ψ p2,b,λ, where θ p 2 p p 2 p 1. We proceed as follows see Fg. 2. Funcon value p 1 p,ψ p Bddng prce p p 2 Fg. 2: An llusraon ψ p,b,λ g p,b,λ Defne a funcon g p,b,λ on p P, where g We have p,b,λ δ p m + w σ +1 σµ r m σµ. ψ p,b,λ g p,b,λ, for all p p p. If r w / σ 2 µ 2, hen g p,b,λ p δ mp Ξ N\{} Ξ 2 N δ mp Ξ N\{} p ψ Ξ 2 N p,b,λ, +m mr µ Ξ N\{} Ξ 2 N +m mw µ Ξ N\{} m µ λ 2 Ξ 2 N for all p < p p. We have p,b,λ g p,b,λ, ψ

7 7 for all p < p p. On he oher hand, accordng o he earler dervaon, we know ha 2 p,b,λ > 0, p 2 g for all p P. Tha s, g p,b,λ s a convex funcon on P, and we oban p,b,λ ψ θ g p1,b,λ + 1 θ g p2,b,λ θψ p1,b,λ + 1 θ g p2,b,λ θ ψ p1,b,λ + 1 θ ψ p2,b,λ. Thus, we have ψ p,b,λ s convex on p P. Ths complees he proof and he resul follows. Theorem 4.2. If boh funcons K 1 x and K 2 x are convex n x X, hen he funcon K 3 x K 1 x+k 2 x s also convex n x X. Proof: A complee proof of he heorem s gven n he supplemenary maeral. Theorem 4.3. There exss a Nash equlbrum soluon se for he formulaed game G P,f, gven ha he condon r w / σ 2 µ 2 N holds. Proof: A complee proof of he heorem s gven n he supplemenary maeral. 4.3 Nash Equlbrum Compuaon Once we have esablshed ha he Nash equlbrum of he formulaed game G P,f exss, we are neresed n obanng a suable algorhm o compue one of hese equlbrums wh mnmum nformaon exchange beween he mulple users and he cloud provders. Noe ha we can furher rewre he opmzaon problem 14 as follows: mnmze f p,ξ N,λ 1 ψ p,ξ N,λ, s.. p,p P, 16 where Ξ N denoes he aggregaed paymens for each server from all cloud users,.e., Ξ N j N p j j. From 16, we can observe ha he calculaon of he dsuly funcon of each ndvdual user only requres he knowledge of he aggregaed paymens for a server from all cloud users Ξ N raher han ha he specfc ndvdual bddng sraegy profle b, whch can brng abou wo advanages. On he one hand, can reduce communcaon raffc beween users and he cloud provder. On he oher hand, can also keep prvacy for each ndvdual user o ceran exen, whch s serously consdered by many cloud users. Snce all users are consdered o be selfsh and ry o mnmze her own dsuly whle gnorng hose of he ohers. I s naural o consder an erave algorhm where, a every eraon k, each ndvdual user N updaes hs/her prce bddng sraegy o mnmze hs/her own dsuly funcon f p,ξ N,λ. The dea s formalzed n Algorhm 1. Algorhm 1 Ierave Algorhm IA Inpu: S, λ S, ǫ. Oupu: p S. 1: //Inalze p for each user S 2: for each cloud user S do 3: se p 0 b. 4: end for 5: Se k 0. 6: //Fnd equlbrum bddng prces 7: whle p k S pk 1 S > ǫ do 8: for each cloud user S do 9: Receve Ξ k S from he cloud provder and compue 10: p k+1 as follows By Algorhm 2: p k+1 argmnf p,ξ k S,λ. p P 11: Send he updaed prce bddng sraegy o he cloud provder. 12: end for 13: Se k k : end whle 15: reurn p k S. Gven S, λ S, and ǫ, where S s he se of cloud users who wan o use he cloud servce, λ S s he reques vecor of all cloud users n S,.e., λ S { } λ S, and ǫ s a relave small consan. The erave algorhm IA fnds opmal bddng prces for all cloud users n S. A he begnnng of he eraons, he bddng prce of each cloud user s se as he conservave bddng prce p. We use a varable k o ndex each of he eraons, whch s nalzed as zero. A he begnnng of he eraon k, each of he cloud users N receves he value Ξ k S from he cloud provder and compues hs/her opmal bddng prce such ha hs/her own dsuly funcon f p,ξ k S,λ S s mnmzed. Then, each of he cloud users n S updaes her prce bddng sraegy and sends he updaed value o he cloud provder. The algorhm ermnaes when he prce bddng sraeges of all cloud users n S are kep unchanged,.e., p k S pk 1 S ǫ. In subsequen analyses, we show ha he above algorhm always converges o a Nash equlbrum f one condon s sasfed for each cloud user. If so, we have an algorhmc ool o compue a Nash equlbrum soluon. Before addressng he convergency problem, we frs presen a propery presened n Theorem 4.4, whch s helpful o derve he convergence resul. } Theorem 4.4. If r > max{ 2δ p w µ, N, hen he σ 2 µ 2 opmal bddng prce p p P of cloud user N s a non-decreasng funcon wh respec o Ξ N\{}, where Ξ N\{} j N p j j p.

8 8 Proof: Accordng o he resuls n Theorem 4.1, we know ha for each cloud user N, gven a fxedb, here are breakpons for he funcon f p,b,λ. We denoe B as he se of he breakpons, hen we have B {p } T, where p m Ξ N\{} λ +σµ Ξ N\{} m m m σµ λ. Combnng he above breakpons wh wo end pons,.e. and p, we oban a new se B { } p, p. Reorder he elemens n B { } 0 p, p such ha p p 1 p p +1, where p 0 p and p +1 p. Then, we oban a new ordered se B. We dscuss he clamed heorem by dsngushng hree cases accordng o he frs dervave resuls of he dsuly funcon f p,b,λ on p P \B. Case 1: p f p,b,λ < 0. Accordng o he resuls n Theorem 4.2, we know ha he second dervave of f p,b,λ on p P \B s posve,.e., 2 p 2 > 0 for all p P \B. In addon, f p,b,λ / f r w σ 2 µ 2, he lef paral dervave s less han ha of he rgh paral dervave n each of he breakpons n B. Therefore, f p f p,b,λ < 0, hen p f p,b,λ < 0 for all p P \B. Namely, f p,b,λ s a decreasng funcon on p P \B. Hence, he opmal bddng prce of cloud user s p p. Tha s o say, he bddng prce of cloud user ncreases wh respec o Ξ. Case 2: p f p,b,λ > 0. Smlar o Case 1, accordng o he resuls n Theorem 4.2, we know ha 2 f p 2 p,b,λ > 0 for all p P \B. Hence, f s an ncreasng p f p,b,λ > 0, f p,b,λ funcon for all p P \B. Therefore, under hs suaon, he opmal bddng prce of cloud user s p p,.e., he opmal bddng prce s always he conservave bddng prce, whch s he nalzed value. Case 3: p f p,b,λ < 0 and p f p,b,λ > 0. Under hs suaon, means ha here exss an opmal bddng prce p P \B such ha f p p,b,λ 1 1 ψ p P p,b,λ +w T r χ 0. p p p 17 Oherwse, he opmal bddng prce for cloud user N s n B. If above equaon holds, hen here exss an neger 0, such ha he opmal bddng prce p s n p,p +1 P \B. Accordng o he dervaons n Theorem 4.1, we know ha he frs dervave of ψ p,b,λ s ψ p,b,λ p mp δ Ξ N\{} +m Ξ 2 N mp δ Ξ N\{} +m Ξ 2 N mrµξ N\{} Ξ 2 < p ; N mwµξ N\{} m µ λ 2 > p Ξ 2, N Tha s, ψ p,b,λ p m Ξ 2 δ p p +2Ξ N\{} r u Ξ N\{} < p N ; m w δ Ξ 2 p p +2Ξ N\{} u Ξ N\{} N m µ λ 2 > p. Therefore, he Eq. 17 s equvalen o he followng equaon: where p,b,λ ϕ hp 1 ϕ p,b,λ 0, δ p p +2Ξ N\{} r u Ξ N\{} < p ; δ p p w +2Ξ N\{} u Ξ N\{} m µ λ 2 > p. Afer some algebrac manpulaon, we can wre he frs dervave resul of ϕ p,b,λ on p as p ϕ p,b,λ 2δ p +Ξ N\{} < p ; 2δ p 2w µ +Ξ N\{} + 2 Ξ2 N\{} m µ λ 3 Ξ 2 > p, N and he frs dervave resul of he funcon ϕ p,b,λ on Ξ N\{} as ϕ p Ξ,b,λ N\{} 2δ p r u < p ; 2δ p r w u µ m µ λ 2 2mwµ2 p Ξ N\{} m µ λ 3 Ξ 2 > p. N Obvously, we have p ϕ p,b,λ > 0, for all p P \B. If r > 2δ p /µ, hen ϕ p Ξ,b,λ < 0. N\{} µ, } w, he funcon hb σ 2 µ 2 Therefore, f r > max{ 2δ p decreases wh he ncrease ofξ N\{}. IfΞ N\{} ncreases, o manan he equaly hb 0, b mus ncrease. Hence, b ncreases wh he ncrease of Ξ N\{}. Ths complees he proof and he resul follows.

9 9 Theorem 4.5. Algorhm IA converges o a Nash} equlbrum, gven ha he condon r > max{ 2δ p w µ, N σ 2 µ 2 holds. Proof: We are now ready o show ha he proposed IA algorhm always converges { o a Nash } equlbrum soluon, gven ha r > 2δ p µ, N holds. w σ 2 µ 2 Le p k be he opmal bddng prce of cloud user N a he k-h eraon. We shall prove above clam by nducon ha p k s non-decreasng n k. In addon, snce p s bounded by p, hs esablshes he resul ha p k always converges. By Algorhm 1, we know ha he bddng prce of each cloud user s nalzed as he conservave bddng prce,.e. 0 s se as p for each of he cloud users N. Therefore, afer he frs eraon, we oban he resuls p 1 p 0 for all N. Ths esablshes our nducon bass. Assumng ha he resul s rue n he k-h eraon,.e. k p k 1 for all N. Then, we need o show ha n he k + 1-h eraon k+1 p k s sasfed for all N. We proceed as follows. By Theorem 4.4, we know ha f r > 2δ p /µ, he opmal bddng prce p of cloud user N ncreases wh he ncrease of Ξ N\{}, where Ξ N\{} j N,j p j j. In addon, we can deduce ha Ξ k N\{} j N,j j N,j p k j j p k 1 j j Ξ k 1 N\{}. Therefore, he opmal bddng prce of cloud user N n he k +1-h eraon p k+1, whch s a funcon of Ξ k N\{}, sasfes pk+1 p k for all N. Thus, he resul follows. Nex, we focus on he calculaon for he opmal bddng prce p n problem 16,.e., calculae p argmnf p,ξ N,λ. 18 p P From Theorem 4.5, we know ha he opmal bddng prce p of cloud user N s eher n B or n P \B such ha f p p,ξ N,λ 1 1 ψ p p,ξ N,λ P δ +w T χ r 0, p p p 19 whereb s an ordered se for all elemens nb { p, p }, and B s he se of breakpons of cloud user N,.e., B {p } T wh p m Ξ N\{} λ +σµ Ξ N\{} m m m σµ λ. 20 Algorhm 2 Calculae p Ξ, λ, ǫ Inpu: Ξ, λ Oupu: p, ǫ.. 1: Se 0. 2: //Fnd p n P \B 3: whle do 4: Se ub p +1 ǫ, and lb p 5: f p f lb,ξ,λ > 0 or hen 6: Se +1; connue. 7: end f 8: whle ub lb > ǫ do +ǫ. p f ub,ξ,λ 9: Se md ub+lb/2, and p md. 10: f p f p,ξ,λ < 0 hen 11: Se lb md. 12: else 13: Se ub md. 14: end f 15: end whle 16: Se p ub+lb/2; break. 17: end whle 18: //Oherwse, fnd p n B 19: f +1 hen 20: Se mn +. 21: for each break pon p 22: f f p,ξ,λ B do < mn hen 23: Se mn f p,ξ,λ 24: end f 25: end for 26: end f 27: reurn p. < 0, and p p. Assumng ha he elemens n B sasfy p0 p 1 p +1, where p 0 p and p +1 p. If equaon 19 holds, hen here exss an neger 0 such ha he opmal bddng prce p p,p +1 P \B. In addon, from he dervaons n Theorem 4.5, we know ha 2 f p,ξ N,λ > 0, 21 p 2 for all p P \B. Therefore, we can use a bnary search mehod o search he opmal bddng prce p n each of he ses p,p +1 P \B 0, whch sasfes 19. If we canno fnd such a bddng prce n P \B, hen he opmal bddng prce p s n B. The dea s formalzed n Algorhm 2. Gven Ξ, λ, and ǫ, where Ξ j N p j j, λ {λ } T, and ǫ s a relavely small consan. Our opmal prce bddng confguraon algorhm o fnd p s gven n Algorhm Calculae p. The key observaon s ha he frs dervave of funcon f p,ξ,λ,.e., p f p,ξ,λ, s an ncreasng funcon n p p,p +1 P \B see 21, where 0. Therefore, f he opmal bddng prce s n P \B, hen we can fnd p by usng he bnary search mehod n

10 10 one of he nervals p,p +1 0 Seps In each of he search nervals p,p +1, we se ub as p +1 ǫ and lb as p +ǫ Sep 4, where ǫ s relave small posve consan. If he frs dervave of funcon f p,ξ,λ on lb s posve or he frs dervave on ub s negave, hen he opmal bddng prce s no n hs nerval Sep 5. Once he nerval, whch conans he opmal bddng prce s decded, we ry o fnd he opmal bddng prce p Seps Noce ha, he opmal bddng prce may n B raher han n P \B Sep 19. Under hs suaon, we check each of he breakpons n B and fnd he opmal bddng prce Seps By Algorhm 2, we noe ha he nner whle loop Seps 8-15 s a bnary search process, whch s very effcen and requres Θ log pmax p ǫ o complee, where p max s he maxmum upper bddng bound of all users,.e., p max max N p. Le max max N, hen he ouer whle loop Seps 3-17 requres me Θ max log pmax p ǫ. On he oher hand, he for loop Seps requres Θ max o fnd soluon n se B. Therefore, he me complexy of Algorhm 2 s Θ max log pmax p ǫ A Near-equlbrum Prce Bddng Algorhm Noce ha, he equlbrum bddng prces obaned by IA algorhm are consdered under he condon ha he allocaed number servers can be fraconal,.e., n he compuaon process, we use nsead of m p j N p j j m, 22 p m j N p m. 23 j j Therefore, we have o revse he soluon and oban a near-equlbrum prce bddng sraegy. Noe ha, under Eq. 23, here may exs some remanng servers, whch s a mos n. Consderng for hs, we reallocae he remanng servers accordng o he bddng prces. The dea s formalzed n our proposed near-equlbrum prce bddng algorhm N PBA, whch characerzes he whole process. A he begnnng, he cloud provder ses a proper conservave bddng prce p and pus s value o no publc nformaon exchange module. Each cloud user N sends hs/her reserved me slos value o he cloud provder. We denoe he curren se of cloud users who wan o use cloud servce as S c and assume ha n he begnnng, all cloud users n N wan o use cloud servce,.e., se S c as N Sep 1. For each curren user se S c, we calculae he opmal bddng prces for all users n S c by IA algorhm, under he assumpon ha he allocaed servers can fraconal Sep 3. And hen, Algorhm 3 N ear-equlbrum Prce Bddng Algorhm N PBA Inpu: N, P, λ N, ǫ. Oupu: p N. 1: Se S c N, S l, and k 0. 2: whle S c S l do 3: Se p N 0, S l S c Sc IAS c,λ Sc,ǫ, and Ξ j N p j j. 4: for each cloud user S c do 5: Compue he allocaed servers as 23,.e., calculae: m p Ξ m. 6: end for 7: Se m R m S c m, and flag rue. 8: whle m R 0 and flag rue do 9: Se flag false. 10: for each cloud user S c do 11: Compue he reallocaed servers,.e., calculae: m p Ξ m R. 12: f u m +m,p,λ > u m,p,λ hen 13: Se m m + m, m R m R m flag false., and 14: end f 15: end for 16: end whle 17: for each cloud user S c do 18: f u m,p,λ < v hen 19: Se p 0, and S c S c {}. 20: end f 21: end for 22: end whle 23: reurn p N. we calculae her correspondng allocaed servers Seps 4-6. We calculae he remanng servers and nroduce a flag varable. The nner whle loop res o allocae he remanng servers accordng o he calculaed bddng sraeges of he curren users n S c Seps The varable flag s used o flag wheher here s a user n S c can mprove hs/her uly by he allocaed number of servers. The whle loop ermnaes unl he remanng servers s zero or here s no one such user can mprove hs/her uly by reallocang he remanng servers. For each user n S c, f hs/her uly value s less han he reserved value, hen we assume ha he/she refuses o use cloud servce Seps The algorhm ermnaes when he users who wan o use cloud servce are kep unchanged Seps PERFORMANCE EVALUATION In hs secon, we provde some numercal resuls o valdae our heorecal analyses and llusrae he performance of he N PBA algorhm. In he followng smulaon resuls, we consder he scenaro conssng of maxmal 200 cloud users. Each me slo s se as one hour of a day and he maxmal me slos of a user can be 72. As shown n TABLE 2, he conservave bddng prce p s vared from 200 o 540 wh ncremen 20. The number of cloud users n s vared from 50 o 200 wh ncremen 10. The maxmal bddng prce p and marke benef facor

11 11 Sysem parameers TABLE 1: Sysem parameers Fxed [Vared range] ncremen Conservave bddng prce p 200 [200, 540] 20 Number of cloud users n 100 [50, 200] 10 Maxmal bddng prce p [500, 800] Marke prof facor r [30, 120] Wegh value w [0.1, 2.5] Reques arrval raes λ [20, 480] Processng rae of a server µ [60, 120] Reservng me slos [1, 72] Reservaon value v 0 Paymen cos wegh δ 1 Oher parameers ǫ, σ, m 0.01, 0.1, 600 r of each cloud user are randomly chosen from 500 o 800 and 30 o 120, respecvely. Each cloud user N chooses a wegh value from 0.1 o 2.5 o balance hs/her me uly and prof. We assume ha he reques arrval rae λ n each me slo of each cloud user s seleced randomly and unformly beween 20 and 480. The processng rae µ of a server o he requess from cloud user N s randomly chosen from 60 o 120. For smplcy, he reservaon value v and paymen cos wegh δ for each of he cloud users are se as zero and one, respecvely. The number of servers m n he cloud provder s se as a consan 600, σ s se as 0.1, and ǫ s se as Bddng prce p User 8 User 18 User 27 User 41 User 59 User Ieraons Fg. 3: Convergence process of bddng prce Fg. 3 shows an nsance for he bddng prces of sx dfferen cloud users versus he number of eraons of he proposed IA algorhm. Specfcally, Fg. 3 presens he bddng prce resuls of 6 randomly seleced cloud users users 8, 18, 27, 41, 59, and 96 wh a scenaro conssng of 100 cloud users. We can observe ha he bddng prces of all users seem o be non-decreasng wh he ncrease of eraon number and fnally reach a relave sable sae, whch verfes he valdness of Theorem 3.4. Tha s, he bddng prces of all cloud users keep unchanged,.e., reach a Nash equlbrum soluon afer several eraons. In addon, can also be seen ha he developed algorhm converges o a Nash equlbrum very quckly. Specfcally, he bddng prce of each user has already acheved a relavely sable sae afer 5 eraon, whch shows he hgh effcency of our developed algorhm. In Fg. 4, we show he rend of he aggregaed paymen from all cloud users P T,.e., he revenue of he cloud provder, versus he ncremen of he conservave bddng prce. We compare wo knds of resuls wh he suaons by compung he allocaed number of servers for each cloud user N as 22 and 23, respecvely. Specfcally, we denoe he obaned paymen asv T when compue m as 22 and P T for 23. Obvously, he former s he opmal value compued from he Nash equlbrum soluon and bgger han ha of he laer. However, canno be appled n a real applcaon, because he allocaed number of servers canno be fraconal. We jus oban a near-equlbrum soluon by assumng ha he allocaed number of servers can be fraconal a frs. Even hough he obaned soluon s no opmal, we can compare hese wo knds of resuls and show ha how closer our proposed algorhm can fnd a near-equlbrum soluon o ha of he compued opmal one. Vrual aggregaed paymen x 10 6 P T V T Value of p Fg. 4: Aggregaed paymen of all users We can observe ha he aggregaed paymen from all cloud users ends o ncrease wh he ncrease of conservave bddng prce a frs. However, decreases when conservave bddng prce exceeds a ceran value. The reason behnd les n ha when conservave bddng prce ncreases, more and more cloud users refuse o use he cloud servce due o he conservave bddng prce exceeds her possble maxmal prce bddng values or her ules are less han her reservaon values,.e., he number of users who choose cloud servce decreases see Fg. 5. We can also observe ha he dfferences beween he values of P T andv T are relavely small and

12 12 Acual number of users Value of p Fg. 5: Acual number of cloud users make lle dfferences wh he ncrease of he conservave bddng prce. Specfcally, he percen dfferences beween he values of V T and P T range from 3.99% o 8.41%, whch reflecs ha our NPBA algorhm can fnd a very well near-opmal soluon whle gnorng he ncremen of conservave bddng prce. To demonsrae hs phenomenon, we furher nvesgae he specfc ules of some users and her correspondng bddng prces, whch are presened n Fg. 6 and Fg. 7. Specfc user uly u 4.5 x User 1 User 19 User 35 User 58 User 87 User Value of p Fg. 6: Specfc user uly In Fg. 6 and Fg. 7, we plo he uly shape and he bddng prces of some cloud users for he developed N PBA algorhm. Fg. 6 presens he uly shape under he developed algorhm versus he ncremen of conservave bddng prce. We randomly selec 6 users users 1, 19, 35, 58, 87, and 100. I can be seen ha he uly rends of all cloud users end o decreases wh he ncrease of conservave bddng prce. However, under every conservave bddng prce, for each user, he dfferences beween he ules compued by usng m as 22 he larger one and 23 he smaller one for each cloud user are relavely small. Therefore, he dfferences beween he aggregaed paymens of P T and V T are small see Fg. 4. Fg. 7 exhbs he cor- an Specfc user bddng prce p User 2 User 19 User 34 User 75 User 87 User Value of p Fg. 7: Specfc User Bddng Prce respondng bddng prces of he users shown n Fg. 6. We can observe ha some users may refuse o use cloud servce when conservave bddng prce exceeds a ceran value user 2. When users choose o use cloud servce, he reads of her bddng prces end o be nondecreasng wh he ncremen of conservave bddng prce user 19, 34, 75, 87, and 100. Ths phenomenon also verfes he aggregaed paymen rend shown n Fg. 4. Specfcally, due o he ncreases of users bddng prces, he aggregaed paymen from all cloud users end o ncrease a frs. However, when conservave bddng prce exceeds a ceran value, more and more cloud users refuse o use cloud servce. Therefore, he aggregaed paymen ends o decrease when conservave bddng prce s large enough. Vrual aggregaed paymen 7 x Number of users n Fg. 8: Aggregaed paymen on number of users In Fg. 8, we show he mpac of number of cloud users on aggregaed paymen. Smlar o Fg. 4, he dfferences beween he values of P T and V T are relavely small. Specfcally, he percen dfferences beween he values of V T and P T range from 3.14% o 12.37%. Tha s, he P T V T

13 13 Acural number of users n an Number of users n Fg. 9: Acural number of cloud users aggregaed paymen resuls for dfferen number of users are largely unchanged. In Fg. 9, we can observe ha wh he ncrease of number of cloud users, he rend of he dfferences beween he number of cloud users and he acual number of cloud users who choose cloud servce also ncreases. The reason behnd les n ha wh he ncrease of number of cloud users, more and more users refuse o use cloud servce due o her ules are less han her conservave values. Ths also parly verfes he aggregaed paymen rend shown n Fg. 8, n whch he aggregaed paymens are largely unchanged wh he ncrease of number cloud users. 6 CONCLUSIONS Wh he popularzaon of cloud compung and s many advanages such as cos-effecveness, flexbly, and scalably, more and more applcaons are moved from local o cloud. However, mos cloud provders do no provde a mechansm n whch he users can confgure bddng prces and decde wheher o use he cloud servce. To remedy hese defcences, we focus on proposng a framework o oban an approprae bddng prce for each cloud user. We consder he problem from a game heorec perspecve and formulae no a non-cooperave game among he mulple cloud users, n whch each cloud user s nformed wh ncomplee nformaon of oher users. For each user, we desgn a uly funcon whch combnes he ne prof wh me effcency and ry o maxmze s value. We desgn a mechansm for he mulple users o evaluae her ules and decde wheher o use he cloud servce. Furhermore, we propose a framework for each cloud user o compue an approprae bddng prce. A he begnnng, by relaxng he condon ha he allocaed number of servers can be fraconal, we prove he exsence of Nash equlbrum soluon se for he formulaed game. Then, we propose an erave algorhm IA, whch s desgned o compue a Nash equlbrum soluon. The convergency of he proposed algorhm s also analyzed and we fnd ha converges o a Nash equlbrum f several condons are sasfed. Fnally, we revse he obaned soluon and propose a near-equlbrum prce bddng algorhm N PBA o characerze he whole process of our proposed framework. The expermenal resuls show ha he obaned near-equlbrum soluon s close o he equlbrum one. As par of fuure drecons, we wll confgure he mulple servers n cloud dynamcally and sudy he relaonshp beween he cloud provder and mulple users. Anoher drecon s o sudy he cloud choce among mulple dfferen cloud provders or deermne a proper mxed bddng sraegy. ACKNOWLEDGMENTS The research was parally funded by he Key Program of Naonal Naural Scence Foundaon of Chna Gran Nos , , he Naonal Naural Scence Foundaon of Chna Gran Nos , , REFERENCES [1] Y. F. B. L and B. L, Prce compeon n an olgopoly marke wh mulple aas cloud provders, Compuers, IEEE Transacons on, vol. 63, no. 1p , Jan [2] R. Pal and P. Hu, Economc models for cloud servce markes: Prcng and capacy plannng, Theorecal Compuer Scence, vol. 496p , [3] P. D. Kaur and I. Chana, A resource elascy framework for qos-aware execuon of cloud applcaons, Fuure Generaon Compuer Sysems, vol. 37p , [4] S. Tang, J. Yuan, C. Wang, and X.-Y. L, A framework for amazon ec2 bddng sraegy under sla consrans, Parallel and Dsrbued Sysems, IEEE Transacons on, vol. 25, no. 1p. 2 11, Jan [5] S. Zaman and D. Grosu, Combnaoral aucon-based allocaon of vrual machne nsances n clouds, Journal of Parallel and Dsrbued Compung, vol. 73, no. 4p , [6] S. Son and K. M. Sm, A prce- and-me-slo-negoaon mechansm for cloud servce reservaons, Sysems, Man, and Cybernecs, Par B: Cybernecs, IEEE Transacons on, vol. 42, no. 3p , June [7] J. Cao, K. Hwang, K. L, and A. Y. Zomaya, Opmal mulserver confguraon for prof maxmzaon n cloud compung, Parallel and Dsrbued Sysems, IEEE Transacons on, vol. 24, no. 6, pp , [8] J. Me, K. L, A. Ouyang, and K. L, A prof maxmzaon scheme wh guaraneed qualy of servce n cloud compung, Compuers, IEEE Transacons on, vol. PP, no. 99p. 1 1, [9] E. Krpeolu, A. en, and K. Gler, Non-cooperave jon replenshmen under asymmerc nformaon, European Journal of Operaonal Research, vol. 227, no. 3p , [10] P. Samm, Y. Temour, and M. Mukhar, A combnaoral double aucon resource allocaon model n cloud compung, Informaon Scences, no. 0p., [11] T. T. Huu and C.-K. Tham, An aucon-based resource allocaon model for green cloud compung, n Cloud Engneerng IC2E, 2013 IEEE Inernaonal Conference on, March 2013p [12] M. Rahman and R. Rahman, Capmaucon: Repuaon ndexed aucon model for resource allocaon n grd compung, n Elecrcal Compuer Engneerng ICECE, h Inernaonal Conference on, Dec 2012p [13] A. Ozer and C. Ozuran, An aucon based mahemacal model and heurscs for resource co-allocaon problem n grds and clouds, n Sof Compung, Compung wh Words and Percepons n Sysem Analyss, Decson and Conrol, ICSCCW Ffh Inernaonal Conference on, Sep 2009p. 1 4.

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Vucec, Auonomous demand sde managemen based on energy consumpon schedulng and nsananeous load bllng: An aggregave game approach, Smar Grd, IEEE Transacons on, vol. 5, no. 4p , Kenl L receved he Ph.D. degree n compuer scence from Huazhong Unversy of Scence and Technology, Chna, n He was a vsng scholar a Unversy of Illnos a Urbana- Champagn from 2004 o He s currenly a full professor of compuer scence and echnology a Hunan Unversy and depuy drecor of Naonal Supercompung Cener n Changsha. Hs major research areas nclude parallel compung, hgh-performance compung, grd and cloud compung. He has publshed more han 130 research papers n nernaonal conferences and journals such as IEEE-TC, IEEE-TPDS, IEEE-TSP, JPDC, ICPP, CCGrd. He s an ousandng member of CCF. He s a member of he IEEE and serves on he edoral board of IEEE Transacons on Compuers. Chubo Lu s currenly workng oward he Ph.D. degree a Hunan Unversy, Chna. Hs research neress are manly n modelng and schedulng of dsrbued compung sysems, approxmaon and randomzed algorhms, game heory, grd and cloud compung. Keqn L s a SUNY Dsngushed Professor of compuer scence. Hs curren research neress nclude parallel compung and hghperformance compung, dsrbued compung, energy-effcen compung and communcaon, heerogeneous compung sysems, cloud compung, bg daa compung, CPU-GPU hybrd and cooperave compung, mulcore compung, sorage and fle sysems, wreless communcaon neworks, sensor neworkseer-o-peer fle sharng sysems, moble compung, servce compung, Inerne of hngs and cyber-physcal sysems. He has publshed over 370 journal arcles, book chapers, and refereed conference papers, and has receved several bes paper awards. He s currenly or has served on he edoral boards of IEEE Transacons on Parallel and Dsrbued Sysems, IEEE Transacons on Compuers, IEEE Transacons on Cloud Compung, Journal of Parallel and Dsrbued Compung. He s an IEEE Fellow. Alber Y. Zomaya s currenly he Char Professor of Hgh Performance Compung and Neworkng and Ausralan Research Councl Professoral Fellow n he School of Informaon Technologes, Unversy of Sydney. He s also he Drecor of he Cenre for Dsrbued and Hgh Performance Compung whch was esablshed n lae Professor Zomaya s he auhor/co-auhor of seven books, more han 400 papers, and he edor of nne books and 11 conference proceedngs. He was he Edor-n- Chef of he IEEE Transacons on Compuers and serves as an assocae edor for 19 leadng journals, such as, he IEEE Transacons on Parallel and Dsrbued Sysems and Journal of Parallel and Dsrbued Compung. Professor Zomaya s he recpen of he Merorous Servce Award n 2000 and he Golden Core Recognon n 2006, boh from he IEEE Compuer Socey. Also, he receved he IEEE Techncal Commee on Parallel Processng Ousandng Servce Award and he IEEE Techncal Commee on Scalable Compung Medal for Excellence n Scalable Compung, boh n Dr. Zomaya s a Charered Engneer, a Fellow of AAAS, IEEE, and IET UK.