An Evolutionary Game Theoretic Approach to Adaptive and Stable Application Deployment in Clouds

Transcription

1 An Evolutonary Game Theoretc Approach to Adaptve and Stable Applcaton Deployment n Clouds Chonho Lee Unversty of Massachusetts, Boston Boston, MA 5, USA chonho@csumbedu Yuj Yamano OGIS Internatonal, Inc San Mateo, CA 9444, USA yyamano@ogsnternatonalcom ABSTRACT Ths paper studes an evolutonary game theoretc mechansm for adaptve and stable applcaton deployment n cloud computng envronments The proposed mechansm, called Nuage, allows applcatons to adapt ther locatons and resource allocaton to the envronmental condtons n a cloud (eg, workload and resource avalablty) wth respect to gven performance objectves such as response tme Moreover, Nuage theoretcally guarantees that every applcaton performs an evolutonarly stable deployment strategy, whch s an equlbrum soluton under gven envronmental condtons Smulaton results verfy ths theoretcal analyss; applcatons seek equlbra to perform adaptve and evolutonarly stable deployment strateges Categores and Subject Descrptors Junch Suzuk Unversty of Massachusetts, Boston Boston, MA 5, USA jxs@csumbedu C [Computer Systems Organzaton]: Computer- Communcaton Networks Dstrbuted Systems; I [Computng Methodologes]: Artfcal Intellgence Keywords Cloud Computng, Adaptve and Stable Applcaton Deployment, Evolutonary Game Theory INTRODUCTION One of key features n cloud computng (eg, Infrastructureas-a-Servce and Platform-as-a-Servce) envronments s elastc scalng of ther applcatons [,, ] In order to provde ths feature, they are requred to perform dynamc applcaton deployment for adjustng the locatons and resource allocaton of cloud applcatons [4, 5, 6, 7] For example, they allocate dfferent amounts of computng and network- Katsuya Oba OGIS Internatonal, Inc San Mateo, CA 9444, USA oba@ogsnternatonalcom Athanasos Vaslakos Unversty of Western Macedona GR 5, Greece vaslako@athforthnetgr ng resources (eg, CPU tme, memory/dsk space and bandwdth) to each applcaton accordng to the applcaton s workload (e, the number of ncomng messages) Ths allows applcatons to operate by balancng dfferent performance objectves such as response tme and resource consumpton (Resource consumpton mples operatonal costs due to the pay-per-use models used n clouds) Moreover, cloud computng envronments relocate an applcaton from one host to another and colocate multple applcatons on the same host accordng to the resource avalablty on hosts Ths allows applcatons to effcently utlze resources and avod the rsk of host crashes due to resource scarcty Ths paper nvestgates two mportant propertes n applcaton deployment n clouds: Adaptablty: allows applcatons to adapt ther locatons and resource allocaton to workload and resource avalablty under gven performance objectves Stablty: allows applcatons to seek stable adaptaton decsons by mnmzng oscllatons (or non-determnstc nconsstences) n decson makng Nuage s an evolutonary game theoretc mechansm for adaptve and stable applcaton deployment n clouds Ths paper descrbes ts desgn and evaluates ts adaptablty and stablty In Nuage, each applcaton contans a set (or a populaton) of multple players that represent dfferent deployment strateges Randomly-pared players repeatedly play games Each game dstngushes a wnnng and a losng player wth respect to performance objectves The wnner replcates tself and ncreases ts share n the populaton The loser s elmnated from the populaton Through multple games performed repeatedly n the populaton, the populaton state (or strategy dstrbuton) changes Through theoretcal analyss, Nuage guarantees that the populaton state converges to an equlbrum where the populaton contans a domnant strategy Nuage performs the domnant deployment strategy as the most ratonal strategy aganst gven workload and resource avalablty Nuage theoretcally proves that the populaton state s evolutonarly stable when t s on an equlbrum An evoluton-

2 arly stable state s the state that, regardless of the ntal populaton state, the populaton state always converges to (A domnant strategy n the evolutonarly stable populaton state s called evolutonarly stable strategy) Thanks to ths property, Nuage guarantees that every applcaton determnstcally performs evolutonarly stable deployment strategy Smulaton results verfy ths theoretcal analyss; applcatons seek equlbra to perform evolutonarly stable deployment strateges and adapt ther locatons and resource allocatons to gven workload and resource avalablty PROBLEM STATEMENT Ths paper consders an applcaton deployment problem where M hosts are avalable to operate N applcatons Each applcaton s desgned and deployed as a set of three server software, followng the three-ter applcaton archtecture [8] Usng a hypervsor such as Xen [9], each server s deployed on a vrtual machne (VM) atop a host The goal of ths problem s to fnd evolutonarly stable deployment strateges that deploy N applcatons (e, N servers) on M hosts so that the applcatons adapt ther locatons and resource allocaton to gven workload and resource avalablty wth respect to performance objectves The placement of and the resource allocaton to each applcaton are conducted on a per-server (or per-vm) bass Ths paper consders CPU tme share (n percentage) as a resource assgned to each server (e, VM) Applcaton Archtecture Each applcaton conssts of the followng three servers: Web server: accepts HTTP messages from applcaton users, valdates data n the messages and provdes Web-based user nterface for users Applcaton server: performs functonal applcaton logc and processes data transmtted from users Database server: takes care of data access and storage Each message s sequentally processed from a Web server to a database server through an applcaton server A reply message s generated by the database server and forwarded n the reverse order toward a user Ths paper assumes that dfferent applcatons utlze dfferent sets of servers (Servers are not shared by dfferent applcatons) Users send dfferent types of messages to dfferent applcatons A host can operate multple VMs, each runs a server Collocated VMs share resources avalable on ther local host Performance Objectves Ths paper consders the followng performance objectves for each applcaton to adapt ts locaton and resource allocaton All objectves are to be mnmzed Response tme: The response tme of an applcaton for ts users It s estmated based on an M/M/ queung model [], n whch message arrvals follow a Posson process and a server s servce tme s exponentally dstrbuted to process ncomng messages Resource consumpton: The total CPU tme share (n percentage) assgned to three vrtual machnes n an applcaton Dstance: The average dstance between VMs n an applcaton It s computed as the hop count between hosts runnng the VMs Load balance: The varance of workload (the number of ncomng messages) among hosts runnng three VMs n an applcaton The response tme of an applcaton (the -th applcaton) s estmated as follows T (s) R = T (s) + T (w) + T (d) () denotes the tme for the -th applcaton to process an ncomng message from a user It s computed as follows T (s),v T (s) = v= T (s),v () denotes the servce tme of the v-th server n the -th applcaton It ndcates how long t takes for the server to process a message denotes the total watng tme for a message to be processed by three servers n the -th applcaton It s computed as follows T (w) T (w) = λ v= ρ,v ρ,v ρ,v () λ denotes the -th applcaton s message arrval rate (e, the number of messages the -th applcaton receves durng the unt tme) ρ,v denotes the utlzaton of the v-th server n the -th applcaton It s computed as follows ρ,v = λ,v C,v/T (s),v v= λ Note that λ =,v (Currently, λ = λ, = λ, = λ,) C,v denotes the CPU tme share allocated to the v-th server n the -th applcaton s the total communcaton delay to transmt a message among servers n the -th applcaton It s obtaned wth the sze of a message and network bandwdth T (d) BACKGROUND: EVOLUTIONARY GAME THEORY Game theory studes strategc selecton of behavors n nteractons among ratonal players In a game, gven a set of strateges, each player strves to fnd a strategy that optmzes ts own payoff dependng on the others strategy choces Game theory seeks such strateges for all players as a soluton, called Nash equlbrum (NE), where no players can gan extra payoff by unlaterally changng hs strategy (4)

3 Evolutonary game theory (EGT) s an applcaton of game theory to bologcal contexts to analyze populaton dynamcs and stablty n bologcal systems In EGT, games are played repeatedly by players randomly drawn from the populaton [, ] In general, EGT consders two major evolutonary mechansms: mutaton, whch njects varetes on genes, and selecton, whch favors some varetes over others based on ther ftness to the envronment Mutaton s consdered n the noton of evolutonarly stable strateges (ESS), whch s a refnement of NE Selecton s consdered n the replcator dynamcs (RD) model Evolutonarly Stable Strateges ESS s a key concept n EGT A populaton followng such a strategy s nvncble Specfcally, suppose that the ntal populaton s programmed to play a certan pure or mxed strategy x (the ncumbent strategy) Then, let a small populaton share of players ɛ (, ) play a dfferent pure or mxed strategy y (the mutant strategy) Hence, f a player s drawn to play the game, the probabltes that ts opponent plays the ncumbent strategy x and the mutant strategy y are ɛ and ɛ, respectvely The player s payoff of such a game s the same as that of a game where the player plays the mxed strategy w = ɛy + ( ɛ)x The payoffs of players wth strateges x and y gven that the opponent adopts strategy w are denoted by U(x, w) and U(y, w), respectvely Defnton A strategy x s called evolutonarly stable f, for every strategy y x, a certan ɛ (, ) exsts, such that the nequalty U(x, ɛy + ( ɛ)x) > U(y, ɛy + ( ɛ)x) (5) holds for all ɛ (, ɛ) In the specal case where the payoff functon s lnear, U(x, w) and U(y, w) can be wrtten as the expected payoffs for players wth strateges x and y, and Equaton (5) yelds ( ɛ)u(x, x) + ɛu(x, y) > ( ɛ)u(y, x) + ɛu(y, y) (6) If ɛ s close to zero, Equaton (6) yelds ether U(x, x) > U(y, x), U(x, x) = U(y, x) and U(x, y) > U(y, y) (7) Hence, t becomes obvous that an ESS must be a NE; otherwse, Equaton (7) do not hold Replcator Dynamcs The replcator dynamcs, frst proposed by Taylor [], specfes how populaton shares assocated wth dfferent pure strateges evolve over tme In replcator dynamcs players are programmed to play only pure strateges To defne the replcator dynamcs, consder a large but fnte populaton of players programmed to play pure strategy k K, where K s the set of strateges At any nstant t, let λ k (t) be the number of players programmed to play pure strategy k The total populaton of players s gven by λ(t) = k K λ k(t) Let x k (t) = λ k (t)/λ(t) be the fracton of players usng pure strategy k at tme t The assocated populaton state s defned by the vector x(t) = [x (t),, x k (t),, x K(t)] Then, the expected payoff of usng pure strategy k gven or that the populaton s n state x s U(k, x) and the populaton average payoff, that s the payoff of a player drawn randomly from the populaton, s U(x, x) = K k= x k U(k, x) Suppose that payoffs are proportonal to the reproducton rate of each player and, furthermore, that a strategy profle s nherted Ths leads to the followng dynamcs for the populaton shares x k ẋ k = x k [U(k, x) U(x, x)] (8) where x k s the tme dervatve of x k The equaton states that populatons wth better (worse) strateges than average grow (shrnk) However, there are cases when even a strctly domnated strategy may gan more than average Hence, t s not a pror clear whether f such strateges get wped out n the replcator dynamcs The followng theorem answers ths queston []: Theorem If a pure strategy k s strctly domnated then ξ k (t, x ) t, where ξ k (t, x )s the populaton at tme t and x s the ntal state On the other hand, t should be noted that the rato x k /x l of two populaton shares x k > and x l > ncreases wth tme f the strctly domnated strategy k gans a hgher payoff than the strctly domnated strategy l Ths s a drect result of Equaton (8) and may be expressed analytcally va d dt [ xk x l ] = [U(k, x) U(l, x)] x k x l (9) From Equaton (9), t s evdent that even suboptmal strateges could temporarly ncrease ther share before beng wped out n the long run However, there s a close connecton between NE and the steady states of the replcator dynamcs, whch s states where the populaton shares do not change ther strateges over tme Thus, snce n NE all strateges have the same average payoff, every NE s a steady state The reverse s not always true: Steady states are not necessarly NE, eg, any state where all players use the same pure strategy s a steady state, but, t s not stable [] In ths paper, a sngle fxed-szed populaton model s used; also, dscrete tme (e, generatonal) model s assumed 4 NUAGE 4 A Desgn of Evolutonary Game n Nuage Nuage executes an ndependent evolutonary game n each of N dfferent applcatons populatons, {AP, AP,, AP N } An applcaton populaton AP contans M players {p,, p,,, p,m } A player has ts own strategy S(p,j) that represents a deployment of three vrtual servers v t correspondng to an applcaton, where t {,, } (eg, for Web, for App, and for DB servers) Each vrtual server s deployed on a host h H The strategy specfes placements and resource allocatons of the vrtual servers A placement ndcates whch host a vrtual server s deployed on An allocaton ndcates CPU tme share assgned to the vrtual server Thus, a strategy s descrbed as a set of pars of the placement and allocaton for three vrtual servers v t where t {,, } and mplemented as S(p,j) = {(h, l ), (h, l ), (h, l )} () where h t denotes a host ID runnng a vrtual server v t for an applcaton, and l t denotes CPU tme share (%) assgned to the vrtual server v t for an applcaton

4 A game s repeatedly performed between randomly pared players n an applcaton populaton Accordng to ther ftness, a wnnng/losng player ncreases/decreases ts subpopulaton n an applcaton populaton A strategy of a player whose subpopulaton s the largest s used as the current deployment of an applcaton Fgure shows example Host strateges for two dfferent applcatons (a and a ) Web Web App App DB DB % % % 5% 45% 5% CPU Tme share Vrtual Vrtual Machnes Machnes for for Applcaton Applcaton a; a; S(a) S(a) = {(, {(, ),(, ),(, ),(, 5),(, 45)} 5)} % % Host Host Fgure : An example allocaton of two applcatons A Nuage state can be descrbed as a deployment state that ndcates an placements and allocatons for all applcatons (e, N vrtual servers) It s denoted as a (N x H )- matrx X = [x rc] where x rc s an allocaton of a vrtual server v r at a host c The vrtual server ID, r, s gven by r = + t where s an applcaton ID, and t s an ndex of vrtual servers {,, } X = 45 N 5 where x rc () r= 5 4 A Procedure of Nuage Applcatons evolve through generatons by changng ther strateges (e, placements and resource allocatons) and mprove ther objectve values (eg, response tme for users) At each generaton, each applcaton repeatedly performs games between randomly-pared players n the applcaton populaton A wnnng player wll make ts copy, and a losng player wll be removed from the populaton Then, a mutaton operaton s performed on each coped player at a certan probablty to change ts strategy The mutaton occurs at one of vrtual servers wth (h t, l t ) One of avalable hosts s randomly assgned to h t, and the value of l t s assgned based on a normal dstrbuton G(µ, σ) = G(l t, ) A player wns/loses aganst the opponent accordng to ther ftness In ths paper, the ftness s gven by a domnaton relatonshp The domnaton relatonshp s determned based on objectves descrbed n Secton and ther prorty By defnton, t s sad that Player A domnates Player B f all objectve values of A s smaller than that of B It s sad that Player A s superor to Player B f the number of domnatng objectves of A s larger than that of B If those numbers for A and B are the same, then they consder the pre-defned prorty of objectves Fgure shows a pseudocode of the mechansm to explan how Nuage works IntalzePopulaton(AP ) ntalzes all applcaton populatons by assgnng randomly chosen strateges to players; Randomze(O) permutates the elements of a set O, whch s a set of ndexes of applcatons; Select(AP ) retreves two players randomly from an applcaton populaton AP ; and PerformGame(p x, p y) determnes wnnng/losng players by evaluatng ther domnaton relatonshp Nuage() // AP: a set of applcaton populatons // O: a permutaton orderng of applcatons ndexes // W: a set of wnnng players, R: a set of the mutated // p x: A player wth a strategy x man IntalzePopulaton(AP ) O (,,, N) whle (the termnaton condton s not satsfed) O Randomze(O) do for r to N do O(r) W, R φ, M AP / for j to M {p, p } Select(AP ) AP AP {p, p } do wnner PerformGame(p, p ) W W wnner R R Mutate(wnner) AP W R Fgure : Evolutonary Games n Nuage 5 STABILITY ANALYSIS Ths secton analyzes the stablty of Nuage by showng that an applcaton populaton state converges to an evolutonarly stable state (or an asymptotcally stable state) n three steps: () The dynamcs of the populaton state change over tme s formalzed as a set of dfferental equatons, () The proposed evolutonary game has equlbrum ponts, () The equlbrum ponts are asymptotcally stable Frst, n order to construct the dfferental equatons, followng termnologes and varables are defned S denotes a set of strateges A strategy, s S, conssts of pars of a placement and allocaton for three vrtual servers as descrbed n Secton 4 S denotes a set of strateges that appear n an applcaton populaton M denotes a populaton sze M = s S n s where n s s the number of players wth a strategy s X(t) denotes a populaton state at tme t X(t) = {x (t), x (t),, x s (t)} where x s s the populaton share of players wth a strategy s (x s = ns ; S s S x s = ) F s s the ftness of a player wth a strategy b p s k denotes the probablty that a player wth a strategy s s replcated by wnnng a game aganst the player wth a strategy k It s computed by p s k = x s φ(f s F k ) where φ(f b F k ) s the condtonal probablty that the ftness of a player wth a strategy s s larger than that of a strategy k How players wth a strategy s change ther populaton share s consdered as the sum of dfference between the number of players whch are replcated (wn) and elmnated (lose)

5 at a tme; then t s formalzed as follows (usng a brevty c sk = φ(f s F k ) φ(f k F s)) ẋ s = {x k p s k x sp k s} k S,k s = x s k S,k s = x s k B,k s x k {φ(f s F k ) φ(f k F s)} x k c sk () Theorem If a player wth a strategy s s strctly domnated, then x s(t) as t In game theory, t s sad that a strategy s strctly domnant f, regardless of what any other players select, a player wth the strategy gans a strctly hgher ftness than any others If a player has a strctly domnant strategy, than t s always better than any others n terms of ftness (e, a domnaton relatonshp) It ncreases ts populaton share and occupes a populaton over tme So, f a player s strctly domnated, then the player dsappears n a populaton over tme Theorem The populaton state of an applcaton populaton converges to an equlbrum Proof It s true that, players wth dfferent strateges have dfferent domnaton factors under the same network condtons In other words, under the partcular network condtons, only one player has the hghest domnaton rank among the others Assume that F > F > > F s, and by Theorem, a populaton state eventually converges to X(t) = {x (t), x (t),, x s (t)} = {,,, } as an equlbrum Dfferental equatons should satsfy the constrant s s x s = Theorem 4 The equlbrum of Nuage s evolutonarly stable (e, asymptotcally stable) Proof At the equlbrum where X = {,,, }, a set of dfferental equatons can be rewrtten n the downszed by substtutng x = x x s ż s = z s[c s( z s) + s =, s where s, k =,, s z c s] () where Z(t) = {z (t), z (t),, z s (t)} denotes the correspondng downszed populaton state, whch s an equlbrum Z eq = {,,, } of ( s )-dmenson based on Theorem To verfy that a state at the equlbrum s an asymptotcally stable state, show that all the Egenvalues of Jaccoban matrx of the downszed populaton state has negatve Real parts The elements of Jaccoban matrx J are [ ] żb J bk = z k Z=Z eq [ zs[c s( z s) + s ] =, s z cs] = (4) z k Z=Z eq where s, k =,, s Therefore, Jaccoban matrx J s gven by c c J = (5) c M where c, c,, c s are the Egenvalues of J Accordng to Theorem, c s = φ(f F s) < for every s; therefore, Z eq = {,,, } s asymptotcally stable 6 EVALUATION Ths secton evaluates the proposed Nuage n smulaton studes and shows how applcatons change ther objectve values over tme by changng ther strateges (e, applcaton deployments) 6 Smulaton Confguratons Ths smulaton consders 5 types of applcatons on 5 hosts (or machnes) whch are fully-connected or lnearly-connected When a dstance objectve s consdered, the lnearly-connected hosts are used Table shows message arrval rate (# of requests/sec) and servce tme (sec) for the applcatons, whch are pre-defned based on Zpf s law [4, 5] The servce tme for applcatons are defned as, 5,, 5, and sec respectvely An entry for the servce tme ndcates the expected processng tme for a sngle message on a vrtual server Table : Message Arrval Rate (# of messages/sec) and Servce Tme (sec) Applcaton type 4 5 Arrval rate 7 4 Web server App server 8 4 DB server Applcaton s assumed to be an applcaton whose computatonal operatons and data management operatons are lght Applcaton s assumed to be an applcaton whose computatonal operatons are heaver than Applcaton Applcaton s assumed to be an applcaton whose data management operatons are heaver than Applcaton Applcaton 4 s assumed to be an applcaton whose operatons s computatonally more ntensve than Applcaton Applcaton 5 s assumed to be an applcaton whose all operatons are heavy For the proposed evolutonary game descrbed n Secton 4, a populaton sze for each applcaton s set to 5, a mutaton probablty s set to, and the communcaton delay T (d) s set to 5 sec All smulaton results are average results of ndependent runs 6 Smulaton Results Smulaton results shows how applcatons strateges (e, deployments) mpact to objectve values Traces of the objectve values are shown n Fgure 4-7 Each subfgure shows traces of objectve values when applcatons consder

6 Table : Objectve Combnatons Combnaton C C C C 4 C 5 C 6 C 7 C 8 Response tme Resource consumpton Dstance Load balancng a partcular combnaton of those objectves The evaluated objectve combnatons are descrbed n Table Frst, n order to nvestgate how applcatons change ther strateges (e, deployments) durng ther evolutonary games, Fgure shows populaton states of applcatons over generatons n a case C 8 Dfferent lnes represent the normalzed number of agents wth dfferent strateges Each applcaton selects a strategy of the agent whose populaton share s the largest n the applcaton populaton (Labeled numbers n the fgure represent strategy IDs) The rght bottom subfgure n Fgure shows a deployment state at generaton n a case C 8 For example, App selects a strategy S 6 = {(, 6)(, 7)(, )} at generaton Fgure 4 shows the average response tme for applcatons The response tme successfully decreases over generatons for applcatons by changng ther deployments Especally, cases C and C result n shorter response tme than the others Placng vrtual servers at the same host contrbutes to reduce response tme In cases C 5, C 7, C 8, the response tme does not decrease as C and C do due to multple objectves In a case C 6, the response tme s relatvely shorter than C 5, C 7, C 8 because t consders a dstance objectve Fgure 5 shows the resource consumpton (e, assgned CPU tme share n %) for applcatons Applcatons try to reduce resource consumpton as much as they can for the effcent use of resources However, n cases C, C, C 4, applcatons does not care for mnmzng resource consumpton and they requre more resource to process ther messages Fgure 6 shows the average dstance (hop counts) among hosts runnng vrtual servers for applcatons These results are evaluated n lnearly-connected hosts Cases C and C result n shorter dstance than the others smlar to the response tme results Placng vrtual servers at the same hosts or closer to each other contrbutes to reduce response tme A case C 6 shows a relatvely shorter dstance compared to C 5, C 7, C 8 because t consders a dstance objectve Fgure 7 evaluates load balancng among hosts runnng vrtual servers for applcatons Load balancng ndex (LBI) s computed as the varance of workload (the number of user messages) among hosts runnng vrtual servers The smaller s the better A case C 4 successfully reduces LBI to around, whch s smaller than the other cases (around -) All the other cases could not reduce LBI as C 4 although C 8 shows a relatvely smaller LBI 7 RELATED WORK Several game theoretc approaches have been proposed for adaptve applcaton placement n clouds [7, 6, 7] They formulate equlbra n applcaton placement and use greedy algorthms to seek equlbrum solutons under multple performance objectves Ths means they do not focus on stablty n applcaton placement In contrast, Nuage employs an evolutonary game theoretc approach and theoretcally guarantees ts stablty n fndng adaptve strateges for resource allocaton as well as applcaton placement under multple performance objectves Nuage also consders mult-ter applcaton archtecture, whle [7, 6, 7] do not [8, 9] and [] seek the optmal placement of a sngle applcaton and mult-server applcatons, respectvely, under a sngle performance objectve It s out of ther scope to seek equlbrum solutons and attan stablty n fndng solutons In contrast, Nuage consders the placement of and the resource allocaton to multple mult-server applcatons It s desgned to fnd stable equlbrum solutons [,, ] propose task/job schedulng mechansms that assgn resources and orders to process tasks, and they assume that the tasks are ndependent each other Those algorthms can stll work wth many loose couplng servce-ntegrated applcatons However, most cloud-based applcatons consst of multple subtasks and requre communcatons among tasks [6] In ths paper, Nuage consders communcatons among vrtual servers where tasks (e, messages) are requred to be sequentally processed on 8 CONCLUSIONS Ths paper descrbes Nuage, an evolutonary game theoretc mechansm for adaptve and stable applcaton deployment n clouds Nuage theoretcally guarantees that every applcaton performs an evolutonarly stable deployment strategy, whch s an equlbrum soluton under gven workload and resource avalablty Smulaton results verfy ths theoretcal analyss; applcatons seek equlbra to perform adaptve and evolutonarly stable deployment strateges As future work, the authors of the paper plan to carry out extended smulaton studes that consder not only CPU tme share but also memory space and network bandwdth as resources It s also planned to compare Nuage wth exstng optmzaton algorthms n order to evaluate the optmalty as well as stablty n Nuage 9 REFERENCES [] A Wess Computng n the clouds ACM networker Magazne, (4), 7 [] J Vara Cloud archtectures Techncal report, Amazon Web Servces, 7 [] G Boss, P Mallad, D Quan, L Legregn, and H Hall Cloud computng Techncal report, IBM Hgh Performance On Demand Solutons, 7 [4] MCarama and SGordan Resource allocaton n grd computng: an economc model Transactons on Computer Research, (), 8 [5] M Stllwell, D Schanzenbach, F Vven, and H Casanova Resource allocaton usng vrtual clusters In Proc of IEEE/ACM Int l Symposum on Cluster Computng and the Grd, May 9 [6] C A Yfouls and A Gounars Honorng SLAs on cloud computng servces: A control perspectve In

7 The normalzed number of agents The normalzed number of agents App S6 S4 S6 8 S9 6 S5 S S89 4 Generaton App4 8 S S57 S7 S S 6 4 Generaton The normalzed number of agents The normalzed number of agents App S75 8 S4 S5 6 4 Generaton App5 S9 8 S4 S6 6 4 Generaton The normalzed number of agents App 8 S67 S S5 S8 6 4 App: VM Generaton VM Host 6 Host Host Host4 6 Host5 App: VM VM App: VM VM VM App4: VM VM VM App5: VM VM VM 5 VM 8 VM Fgure : Populaton states of applcatons and the deployment state at generaton ( means ) Proc of EUCA/IEEE European Control Conference, August 9 [7] S U Khan and C Ardl Energy effcent resource allocaton n dstrbuted computng systems In Proc of WASET Int l Conference on Dstrbuted, Hgh-Performance and Grd Computng, August 9 [8] B Urgaonkar, G Pacfc, P Shenoy, M Spretzer, and A Tantaw An analytcal model for mult-ter nternet servces and ts applcatons In Proc of ACM Int l Conference on Measurement and Modelng of Computer Systems, June 5 [9] P Barham, B Dragovc, K Fraser, S Hand, T Harrs, A Ho, R Neugebauer, I Pratt, and A Warfeld Xen and the art of vrtualzaton In Proc of ACM symposum on operatng systems prncples, October [] R B Cooper Introducton to Queueng Theory North-Holland Elsever, 98 [] J W Webull Evolutonary Game Theory MIT Press, 996 [] M A Nowak Evolutonary Dynamcs: Explorng the Equatons of Lfe Harvard Unversty Press, 6 [] P Taylor and L Jonker Evolutonary stable strateges and game dynamcs Elsever Mathematcal Boscences, 4(), 978 [4] R Perlne Zpf s law, the central lmt theorem, and the random dvson of the unt nterval Physcal Revew E, 54(), 996 [5] J Tatemura, W-P Hsung, and W-S L Acceleraton of web servce workflow executon through edge computng In Proc of Int l World Wde Web Conference, May [6] G We, A V Vaslakos, Y Zheng, and N Xong A game-theoretc method of far resource allocaton for cloud computng servces Journal of Supercomputng, Accepted, 9 [7] N Doulams, A Doulams, A Ltke, A Panagaks, T Varvargou, and E Varvargos Adjusted far schedulng and non-lnear workload predcton for qos guarantees n grd computng Elsever Computer Communcatons, (), 7 [8] C S Xaoyun, X Zhu, and H Crowder A mathematcal optmzaton approach for resource allocaton n large scale data centers Techncal report, HP Labs Palo Alto, [9] A Zhang, P Santos, D Beyer, and H k Tang Optmal server resource allocaton usng an open queung network model of response tme Techncal report, HP Labs Palo Alto, [] S Borst, O Boxma, J F Groote, and S Mauw Task allocaton n a mult-server system Journal of Schedulng, 6(5), [] R P Doyle Model-based resource provsonng n a web servce utlty In Proc of USENIX Symposum on Internet Technologes and Systems, March [] R Buyya, D Abramson, J Gddy, and H Stocknger Economc models for resource management and schedulng n grd computng Journal of Concurrency and Computaton, 4(), [] A Byde, A Byde, M Salle, M Salle, and C Bartoln Market-based resource allocaton for utlty data centers Techncal report, HP Labs Brstol,

8 5 5 5 C C C C C5 C6 4 C7 4 C Fgure 4: response tme for users (sec) over generatons (x-axs) C C 5 C C4 5 C5 C6 C7 C Fgure 5: Resource consumpton (%) over generatons (x-axs) C C C C C5 5 4 C C C Fgure 6: dstance among hosts runnng vrtual servers (hop counts) over generatons (x-axs) C C C 5 C C5 5 C6 5 C7 C Fgure 7: Load balancng ndex (LBI) over generatons (x-axs)