Simple and Effective Dynamic Provisioning for Power-Proportional Data Centers

Transcription

1 Simpl and Effctiv Dynamic Provisioning for Powr-Proportional Data Cntrs Tan Lu, Minghua Chn, and Lachlan L. H. Andrw Abstract Enrgy consumption rprsnts a significant cost in data cntr opration. A larg fraction of th nrgy, howvr, is usd to powr idl srvrs whn th workload is low. Dynamic provisioning tchniqus aim at saving this portion of th nrgy, by turning off unncssary srvrs. In this papr, w xplor how much gain knowing futur workload information can bring to dynamic provisioning. In particular, w dvlop onlin dynamic provisioning solutions with and without futur workload information availabl. W first rval an lgant structur of th off-lin dynamic provisioning problm, which allows us to charactriz th optimal solution in a divid-and-conqur mannr. W thn xploit this insight to dsign two onlin algorithms with comptitiv ratios 2 α and / + α), rspctivly, whr 0 α is th normalizd siz of a look-ahad window in which futur workload information is availabl. A fundamntal obsrvation is that futur workload information byond th fullsiz look-ahad window corrsponding to α = ) will not improv dynamic provisioning prformanc. Our algorithms ar dcntralizd and asy to implmnt. W dmonstrat thir ffctivnss in simulations using ral-world tracs. I. Introduction Cloud computing is a nw paradigm for providing Intrnt srvics to a larg volum of nd-usrs. In this paradigm, cloud computing srvic providrs provid infrastructur, in particular data cntrs, as a srvic and charg customrs basd on thir usag. Howvr, th nrgy consumption of data cntrs hosting ths srvics has bn skyrockting. In 200, data cntrs worldwid consumd an stimatd 240 billion kilowatt-hours kwh) of nrgy, roughly.3% of th world total nrgy consumption [2]. Powr consumption at such a lvl is almost nough to powr all of Spain [3]. Enrgy-rlatd costs ar approaching th cost of IT hardwar in data cntrs [4], and ar growing 2% annually [5]. Rcnt work has xplord lctricity pric fluctuation in tim and gographically balancing load across cloud data cntrs to cut th lctricity costs; s.g., [6], [7], [8], [9] and th rfrncs thrin. To bnfit from this, th nrgy consumption of a data cntr must rflct its actual load. Enrgy consumption in a data cntr is a product of th powr usag ffctivnss PUE) and th nrgy consumd by th srvrs. Thr hav bn substantial fforts in improving PUE,.g., by optimizing cooling [0], [] and powr managmnt [2]. In this papr, w focus on rducing th nrgy consumd by th srvrs. A prliminary vrsion of th papr appard in CISS in 202[]. Tan Lu and Minghua Chn ar with Dpartmnt of Information Enginring, Th Chins Univrsity of Hong Kong, Hong Kong. Lachlan L. H. Andrw is with th Cntr for Advancd Intrnt Architcturs, Swinburn Univrsity of Tchnology, Australia. PUE is dfind as th ratio btwn th amount of powr ntring a data cntr and th powr usd to run its computr infrastructur. Th closr to on PUE is, th bttr nrgy utilization is. Ral-world statistics rvals thr obsrvations that suggst ampl saving is possibl in srvr nrgy consumption [3], [4], [5], [6], [7], [8]. First, workload in a data cntr oftn fluctuats significantly on th timscal of hours or days, xprssing a larg pak-to-man ratio. Scond, data cntrs today oftn provision for far mor than th obsrvd pak to accommodat both th prdictabl workload and th unprdictabl flash crowds. Such static ovr-provisioning rsults in low avrag utilization for most srvrs. Third, a lightly-utilizd or idl srvr consums mor than 60% of its pak powr. Ths obsrvations imply that a larg portion of th nrgy consumd by srvrs gos into powring narly-idl srvrs, and it can b bst savd by turning off srvrs during th off-pak priods. In particular, an important tchniqu for rducing th nrgy consumption of idl srvrs is for srvrs to autonomously turn off sub-systms [9]. On promising tchniqu xploiting th abov insights is dynamic provisioning, which turns on a minimum numbr of srvrs to mt th currnt dmand and dispatchs th load among th running srvrs to mt Srvic Lvl Agrmnts SLA), making th data cntr powr-proportional. This is nabld by virtualization, which is th fundamntal tchnology that allows th cloud to xist. Thr has bn a significant amount of ffort in dvloping such tchniqu, initiatd by th pionring works [3], [4] a dcad ago. Among thm, on lin of work [9], [6], [5] xamins th practical fasibility and advantag of dynamic provisioning using ral-world tracs, suggsting substantial gain is indd possibl in practic. Anothr lin of work [3], [20], [5] focuss on dvloping algorithms by utilizing various tools from quuing thory, control thory and machin larning to provid insights that can lad to ffctiv solutions. Ths xisting works provid a numbr of schms that dlivr favorabl prformanc ustifid by thortic analysis and/or practical valuations. S [2] for a rcnt survy. Howvr, turning srvrs on and off incurs a cost. Hnc th ffctivnss of ths xciting schms usually rlis on th ability to prdict futur workload to a crtain xtnt,.g., using modl fitting to forcast futur workload from historical data [5]. This naturally lads to th following qustions: Can w dsign onlin solutions that rquir zro futur workload information, yt still achiv clos-to-optimal prformanc? Can w charactriz th bnfit of knowing futur workload in dynamic provisioning? Answrs to ths qustions provid fundamntal undrstanding on how much prformanc gain on can hav by xploiting futur workload information in dynamic provisioning.

2 2 Th worst-cas prformanc of an on-lin algorithm A is oftn masurd by its comptitiv ratio: th maximum, ovr all possibl problm instancs, of ratio of th cost of th solution found by A to th cost of th optimal off-lin solution. Rcntly, Lin t al. [22] proposd an algorithm that rquirs almost-zro futur workload information 2 and achivs a comptitiv ratio of 3, i.., th nrgy consumption is at most 3 tims th minimum computd with prfct futur knowldg). In simulations, thy furthr show th algorithm can xploit availabl futur workload information to improv th prformanc. Ths rsults ar vry ncouraging, indicating that a complt answr to th qustions is possibl. In this papr, w furthr xplor answrs to th qustions, and mak th following contributions: W considr a scnario whr a running srvr consums a fixd amount of nrgy pr unit tim. W rval that th dynamic provisioning problm has an lgant structur that allows us to solv it in a divid-and-conqur mannr. This insight lads to a full charactrization of th optimal solution, achivd by a cntralizd procdur. W show that th optimal solution can also b attaind by a simpl last-mpty-srvr-first ob-dispatching stratgy and ach srvr indpndntly solving a classic skirntal problm. W build upon this architctural insight to dsign two dcntralizd onlin algorithms. Th first, namd CSR, is dtrministic with comptitiv ratio 2 α, whr 0 α is th normalizd siz of a lookahad window in which futur workload information is availabl. Th scond, namd RCSR, is randomizd with comptitiv ratio / + α). W prov that 2 α and / + α) ar th bst comptitiv ratios for dtrministic and randomizd onlin algorithms undr last-mpty-srvr-first ob-dispatching stratgy. Our rsults lad to a fundamntal obsrvation: undr th cost modl that a running srvr consums a fixd amount of nrgy pr unit tim, futur workload information byond th full-siz look-ahad window will not improv th dynamic provisioning prformanc. Th siz of th full-siz look-ahad window is dtrmind by th warand-tar cost and th unit-tim nrgy cost of on srvr. W also xtnd th algorithms to th cas whr srvrs tak stup tim T s to turn on and workload a t) satisfis aτ) + γ)at) for all τ [t, t + T s ], achiving comptitiv ratios uppr boundd by 2 α) + γ) + 2γ +α and + γ) + 2γ. Our algorithms ar simpl and asy to implmnt. W dmonstrat th ffctivnss of our algorithms in simulations using ral-world tracs. W also compar thir prformanc with stat-of-th-art solutions. Th rst of th papr is organizd as follows. W formulat th problm in Sction II. Sction III rvals th important structur of th formulatd problm, charactrizs th optimal solution, and dsigns a simpl dcntralizd offlin algorithm achiving th optimal. In Sction IV, w propos onlin algorithms and provid prformanc guarants. Sction V 2 LCP algorithm [22] is a discrt tim algorithm that only rquirs an stimat of th ob arrival rat of th currnt slot. prsnts th xprimnts and Sction VI concluds th papr. A. Sttings and Modls II. Problm Formulation W considr a data cntr consisting of a st of homognous srvrs. Without loss of gnrality, w assum ach srvr has a unit srvic capacity 3, i.., it can only srv on unit workload pr unit tim. Lt th unit tim powr consumption of busy and idl srvrs b P b and P, rspctivly. W dfin β on and β o f f as th cost of turning a srvr on and off, rspctivly. This includs war-and-tar costs, including th amortizd srvic intrruption cost and procurmnt and rplacmnt cost of srvr componnts hard-disks and powr supplis in particular). [20], [23]. It is comparabl to th nrgy cost of running a srvr for svral hours [22]. Th rsults w dvlop in this papr apply to both of th following two typs of workload: mic typ workloads, such as rqust-rspons wb srving. Each ob of this typ has a small transaction siz and short duration. A numbr of xisting works [3], [4], [22], [24] modl such workloads by a discrt-tim fluid modl. In th modl, tim is dividd into qual-lngth slots. Jobs arriving in on slot gt srvd in th sam slot. Workload can b split among running srvrs at arbitrary granularity lik a fluid. lphant typ workloads, such as virtual machin hosting in cloud computing. Each ob of this typ has a larg transaction siz, and can last for a long tim. W modl such workload by a continuous-tim brick modl. In this modl, tim is continuous, and w assum on srvr can only srv on ob 4. Jobs arriv and dpart at arbitrary tims, and no two ob arrival/dpartur vnts happn simultanously. For th discrt-tim fluid modl, srvrs toggld at th discrt tim poch will not intrrupt ob xcution and thus no ob migration is incurrd. This nat abstraction allows rsarch to focus on srvr on-off schduling to minimiz th cost. For th continuous-tim brick modl, whn a srvr is turnd off, th long-lasting ob running on it nds to b migratd to anothr srvr. In gnral, such non-trivial migration cost nds to b takn into account whn toggling srvrs. In th following, w prsnt our rsults basd on th continuous-tim brick modl. W add discussions to show th algorithms ar also applicabl to th discrt-tim fluid modl. W assum that ach ob is prsnt on a closd intrval of tim. Th numbr of obs as a function of tim is thn a nonngativ, intgr valud uppr smi-continuous function a. For convninc, w furthr assum that a changs by at most 3 In practic, srvr s srvic capacity can b dtrmind from th kn of its throughput and rspons-tim curv [6]. 4 This could b ustifid if thr wr a SLA in cloud computing that rquirs th ob dos not shar th physical srvr with othr obs du to scurity concrns. Th problm is substantially diffrnt if a singl srvr can host multipl virtual machins VMs). Spcifically, if th schduling disciplin is rstrictd to bing non-clairvoyant ob sizs ar only known whn thy complt) thn VM migration bcoms much mor bnficial than in th cas that schduling disciplin is clairvoyant; without VM migration, th comptitiv ratio is at last as larg as th numbr of VMs that can b hostd on a singl srvr in th cas that schduling disciplin is non-clairvoyant.

3 3 at any tim. To avoid tchnicalitis, w assum a is boundd and not always zro. Th numbr of srvrs on srving or idl) can b dfind as follows. For ach srvr s, dfin a function u s that is right continuous with u s 0 ) = 0, counting th numbr of tims th srvr has turn on, and a function d s that is lft continuous with and d s 0) = 0, counting th numbr of tims th srvr has turnd off. Th stat of srvr s at tim t is thn x s t) = u s t) d s t), which must b ithr 0 or. Thn dfin u = s u s and d = s d s. Th total numbr of srvrs on is x = u d. To focus on th cost within [0, T], w rquir x0) = a 0) and x T) = a T). For convninc, w st a t) = 0 for all t < 0 and all t > T. Dfin th cost of srvr s on an intrval [t, t 2 ) as t 2 c s t, t 2 ) = P x s t) dt+β on u s t 2 ) u s t ))+β o f f d s t 2 ) d s t )) t ) whr th intgral rprsnts th running cost, and th othr trms ar th switching costs. Not that this includs any cost of switching on at t 2 vn though that is not in th intrval, and nglcts th cost of switching off at t vn though that is in th intrval. Consquntly, for any t < t 2 < t 3 w hav c s t, t 3 ) = c s t, t 2 ) + c s t 2, t 3 ). It will somtims b usful to considr th ntir switching cost on a closd intrval. Lt P on t, t 2 ) and P o f f t, t 2 ) dnot th total war-and-tar cost incurrd by turning on and off srvrs in [t, t 2 ], rspctivly. Thy tak th on-off cost at t and t 2 into account. Spcifically, if u has lft limits and d has right limits, thn P on t, t 2 ) = β on u s t 2 ) u s t )) and P o f f t, t 2 ) = β o f f d s t 2 + ) d st )). Our rsults dpnd only on th sum P on + P o f f, but w rtain both trms to mphasiz th two physical procsss. B. Problm Formulation W formulat th problm of minimizing srvr opration cost in a data cntr in an intrval [T, T 2 ] givn an initial numbr of on srvrs X and a final numbr of on srvrs X 2 as follows: x t) dt + P on T, T 2 ) + P o f f T, T 2 ) 2) P[a t), X, X 2, T, T 2 ] : T 2 min P T s.t. xt) at), t [T, T 2 ], 3) xt ) = X, xt 2 ) = X 2, 4) var xt) Z +, t [T, T 2 ], 5) whr Z + dnots th st of non-ngativ intgrs. In particular, w ar intrstd in th Srvr Capacity Provisioning problm, SCP, givn by P[a t), a0), at), 0, T]. Th obctiv is to minimiz th sum of srvr nrgy consumption and th war-and-tar cost. Th actual summation of th two parts of th cost is T 0 P [x t) a t)] + P ba t) dt + P on 0, T)+P o f f 0, T), sinc th busy and idl powrs can diffr. Howvr T 0 P ba t) Pa t) dt is constant for givn a t), and so to minimiz th total cost is to minimiz 2). Th constraints in 3) say th srvic capacity must satisfy th dmand. Th constraints in 4) ar th boundary conditions. Rmarks: ) Th problm SCP dos not considr th possibl migration cost associatd with th continuous-tim discrtload modl. Fortunatly, our rsults latr show that w can schdul srvrs according to th optimal solution, and at th sam tim dispatch obs to srvrs in a way that aligns with thir on-off schduls, thus incurring no migration cost. Hnc, th minimum srvr opration cost rmains unaltrd vn w considr migration cost in th problm SCP which can b rathr complicatd to modl). 2) Th formulation rmains th sam with th discrt-tim fluid workload modl whr thr is no ob migration cost to considr. 3) Th problm SCP is similar to a common on considrd in th litratur,.g., in [22], with a spcific linar) cost function. Th bnfit of SCP is that w rtain th constraint that th dcision variabls b intgrs instad of ral numbrs. This is important for clustrs and small data cntrs. Thr ar an infinit numbr of intgr variabls x t), t [0, T], in th problm SCP, which mak it challnging to solv. Morovr, in practic th data cntr has to solv th problm without knowing th workload at), t [0, T] ahad of tim. In rality, a t) is not continuous and it may b right continuous or lft continuous at all th discontinuous points. Howvr, in our SCP problm, w will modify a t) to mak it right continuous whn a t) incrass by on lft continuous whn at) dcrass by on. This simpl modification will not chang th optimal valu of SCP. Nxt, w dsign an off-lin algorithm, including i) a obdispatching algorithm and ii) a srvr on-off schduling algorithm, to solv th problm SCP optimally. W thn xtnd th algorithm to its on-lin vrsions and analyz thir prformanc guarants with or without partial) futur workload information. III. Optimal Solution and Offlin Algorithm W study th off-lin vrsion of th srvr cost minimization problm SCP, whr th workload at) in [0, T] is givn. W first dsign a procdur to construct an optimal solution to problm SCP. W thn driv a simpl and dcntralizd algorithm, upon which w build our onlin algorithms. A. Structur of Optimal Solution W first dfin th critical intrval as β on + β o f f 6) P Lt M b th maximum valu of at), t [0, T]. W thn dfin ā t),t [ 2, T + 2 ] as an xtnsion of a t): a t) t [0, T] ā t) = 0 t 2, 0) T, T + 2 ) M + t { 2, T + 2 }

4 4 at) at) x t) at) 0 T T T 2 T 3 T 4 t 2 T 0 T T 2 T t Figur : Exampl of solution constructd by Optimal Solution Construction Procdur. 2 3 Lt x t), t [0, T], b an optimal solution to th problm SCP, and th corrsponding minimum srvr opration cost b P. Th optimal solution can b constructd as follows. Optimal Solution Construction Procdur: For A from to M + do Find all th intrvals τ, τ )in [ 2, T + 2 ] such that ā τ) A, ā τ ) A and ā t) < A, t τ, τ ). For all intrvals τ, τ ) do If τ τ thn r)assign x t) min [ā τ), ā τ )]; Els for any part of th intrval that x t) has not alrady bn st, st x t) ā t). End if End For End For On xampl of x t), t [0, T] can b found in Fig.. Th following thorm is provd in Appndix A using proof-by-contradiction and counting argumnts. Thorm. Th rsult of th Optimal Solution Construction Procdur, x t), t [0, T], is an optimal solution to th problm SCP. Morovr, th optimal u s and d s hav both lft and right limits. B. Intuitions and Obsrvations Considr th xampl shown in Fig. 2. During [0, T], th systm starts and nds with two obs and two running srvrs. Lt th srvrs whos obs lav at tims 0 and T b S and S2, rspctivly. At tim 0, a ob lavs. Lt T b th tim T until at) again rachs th lvl a0). Th procdur compars T against. If > T, thn it sts xt) = 2 and kps all two srvrs running for all t [0, T]; othrwis, according to Optimal Solution Construction Procdur, xt) = for t [0, T ] [T 2, T] and xt) = 0, t [T, T 2 ] if > δ or xt) = 0, t [T, T 2 ] if δ. Ths actions rval two important obsrvations, upon which w build a dcntralizd off-lin algorithm to solv th problm SCP optimally. Nwly arrivd obs should b assignd to srvrs in th rvrs ordr of thir last-mpty-pochs. Upon bing assignd an mpty priod, a srvr only nds to indpndntly mak locally nrgy-optimal dcision Figur 2: An xampl of a tim priod [0, T]. Intrval δ = T 2 T, δ 2 = T 2, and δ 3 = T T. In th xampl, whn a nw ob arrivs at tim T 2, th procdur implicitly assigns it to srvr S2 instad of S. As a rsult, S and S2 hav mpty priods of T and δ, rspctivly. This may sound unfair compard to an altrnativ stratgy that assigns th ob to th arly-mptid srvr S, which givs S and S2 mpty priods of δ 2 and δ 3, rspctivly. Howvr, at ach dcision point, allocating to th last-mpty srvr rsults in a distribution of th idl tims I that is convxly largr than that rsulting from any othr allocation; i.., it maximizs E[I x) + ] for all x [25]. Not that if x is th tim aftr which th srvr dcids to slp, thn E[I x) + ] is th xpctd nrgy saving. It is straightforward to vrify that in th xampl, upon a ob laving srvr S at tim 0, th procdur implicitly assigns an mpty-priod of T to S, and turns S off if < T and kps it running at idl stat othrwis. Similarly, upon a ob laving S2 at tim T, S2 is turnd off if < δ and stays idl othrwis. Such comparisons and dcisions can b don by individual srvrs thmslvs. C. Offlin Algorithm Achiving th Optimal Solution Th Optimal Solution Construction Procdur dtrmins how many running srvrs to maintain at tim t, i.., x t), to achiv th optimal srvr opration cost P. Howvr, as discussd in Sction II-A, undr th continuous-tim brick modl, schduling srvrs on/off according to x t) might incur non-trivial ob migration cost. Exploiting th two obsrvations mad in th cas-study at th nd of last subsction, w dsign a simpl and dcntralizd off-lin algorithm that givs an optimal x t) and incurs no ob migration cost. Dcntralizd Off-lin Algorithm: A cntral ob-dispatching ntity implmnts a last-mptysrvr-first stratgy. In particular, it maintains a stack i.., a Last-In/First-Out quu) storing th IDs for all idl or off srvrs. Bfor tim 0, th stack contains idntifirs IDs) for all th srvrs that ar not srving. Upon a ob arrival: th dispatchr pops a srvr ID from th top of th stack, and assigns th ob to th corrsponding srvr if th srvr is off, th dispatchr turns it on).

5 5 Upon a ob dpartur: a srvr bcoms idl, and th dispatchr pushs th srvr ID into th stack. Each srvr: Upon rciving a ob: th srvr starts srving th ob immdiatly. Upon a ob laving this srvr: lt th dpartur poch b t. Th srvr sarchs for th arlist tim t 2 t, t + ] such that at 2 ) = at ). If no such t 2 xists, thn th srvr turns itslf off. Othrwis, it stays idl. W rmark that in th algorithm, w us th sam srvr to srv a ob during its ntir soourn tim. Thus thr is no ob migration cost. Th following thorm ustifis th optimality of th off-lin algorithm. Thorm 2. Th proposd off-lin algorithm achivs th optimal cost of th problm SCP. Proof: Rfr to Appndix B. Thr ar two important obsrvations. First, th obdispatching stratgy only dpnds on th past ob arrivals and dparturs. Consquntly, th stratgy assigns a ob to th sam srvr no mattr whthr it knows futur ob arrival/dpartur tims or not; it also acts indpndntly from srvrs off-or-idl dcisions. Scond, ach individual srvr is actually solving a classic ski-rntal rnt-or-buy) problm [26]. Each chooss whthr to rnt kp idl and pay an ongoing nrgy cost) or to buy turn off now and pay a onoff war-and-tar cost), but th numbr of rounds is ointly dtrmind by th ob-dispatching stratgy. Nxt, w xploit ths two obsrvations to xtnd th off-lin algorithm to its onlin vrsions with prformanc guarants. IV. Onlin Dynamic Provisioning with or without Futur Workload Information Inspird by our off-lin algorithm, w construct onlin algorithms by combining i) th sam last-mpty-srvr-first ob-dispatching stratgy as th on in th proposd off-lin algorithm, and ii) an off-or-idl dcision modul running on ach srvr to solv an onlin ski-rntal problm. To valuat our onlin algorithms, w compar its prformanc to that of th bst off-lin algorithm. W say a dtrministic onlin algorithm A is R-comptitiv if for all input squncs σ, w hav C A σ) RC opt σ) + O) whr C A σ) is th cost of algorithm A and C opt σ) is th offlin optimal. W say a randomizd onlin algorithm A, is R-comptitiv 5 if for all input squncs σ, w hav E [C A σ)] RC opt σ) + O) whr E [C A σ)] is th xpctation of th cost of algorithm A with rspct to its random choics for input squnc σ, and C opt σ) is th offlin optimal. 5 against an oblivious advrsary As discussd at th nd of last sction, th last-mptysrvr-first ob-dispatching stratgy utilizs only past ob arrival/dpartur information. Consquntly in both th offlin and onlin cass, it assigns th sam st of obs to th sam srvr at th sam squnc of pochs. Lmma 3. For th sam a t), t [0, T], undr th last-mptysrvr-first ob-dispatching stratgy, ach srvr will gt th sam ob at th sam tim and th ob will lav th srvr at th sam tim for both off-lin and onlin situations. Lmma 3 is tru bcaus last-mpty-srvr-first obdispatching stratgy only dpnds on past workload and it is indpndnt of th past statuss of srvrs. As a rsult, in th onlin cas, ach srvr still facs th sam st of off-or-idl problms as in th off-lin cas. This is th ky to driving th comptitiv ratios of th onlin algorithms w will prsnt. Each srvr, not knowing th mpty priods ahad of tim, nds to dcid whthr to stay idl or turn off and if so whn) in an onlin fashion. On natural approach is to adopt classic algorithms for th onlin ski-rntal problm. A. Dynamic Provisioning without Futur Workload Information For th onlin ski-rntal problm, th brak-vn algorithm in [26] and th randomizd algorithm in [27] hav comptitiv ratios 2 and / ), rspctivly. Th ratios hav bn provd to b optimal for dtrministic and randomizd algorithms, rspctivly. Dirctly adopting ths algorithms in th off-or-idl dcision modul lads to two onlin solutions for th problm SCP with comptitiv ratios 2 and / ).58. Ths ratios improv th bst known ratio 3 achivd by th algorithm in [22]. Th rsulting solutions ar dcntralizd and asy to implmnt: a cntral ntity runs th last-mpty-srvr-first obdispatching stratgy, and ach srvr indpndntly runs an onlin ski-rntal algorithm. For xampl, if th brak-vn algorithm is usd, a srvr that ust bcoms mpty at tim t will stay idl for an amount of tim. If it rcivs no ob during this priod, it turns itslf off. Othrwis, it starts to srv th ob immdiatly. As a spcial cas covrd by Thorm 5, it turns out this dirctly givs a 2-comptitiv dynamic provisioning solution. B. Dynamic Provisioning with Futur Workload Information Studis of onlin problms usually assum zro futur information. Howvr, in our data cntr dynamic provisioning problm, on ky obsrvation many xisting solutions xploitd is that th workload xhibits highly rgular pattrns. Thus th workload information in a nar look-ahad window may b accuratly stimatd by machin larning or modl fitting basd on historical data [5], [28]. Can w xploit such futur knowldg, if availabl, in dsigning onlin algorithms? If so, how much gain can w gt? Lt us laborat through an xampl to xplain why and how much futur knowldg can hlp. Suppos at any tim t, th workload information at) in a look-ahad window [t, t + α ]

6 6 is availabl, whr α [0, ] is a constant. Considr a srvr running th brak-vn algorithm that bcoms mpty at tim t, and has an mpty priod infinitsimally longr than. Following th standard brak-vn algorithm, th srvr waits for tim bfor turning itslf off. In this xampl, it rcivs a ob immdiatly aftr t + poch, and it has to powr up to srv th ob. This incurs a total cost of 2P as compard to th optimal on P, which is achivd by th srvr staying idl all th way. Anothr stratgy that costs lss is as follows. Th srvr stays idl for an amount of tim α), and pks into th look-ahad window [t + α), t + ]. Du to th lastmpty-srvr-first ob-dispatching stratgy, th srvr can asy tll that it will rciv a ob if any at) in th window xcds at ), and no ob othrwis. In this xampl, th srvr ss itslf rciving no ob during [t + α), t + ] and it turns itslf off at tim t + α). Latr it turns itslf on to srv th ob right aftr t +. Undr this stratgy, th ovrall cost is 2 α) P, which is bttr than that of th brak-vn algorithm. This xampl shows it is possibl to modify classic onlin algorithms to xploit futur workload information to obtain bttr prformanc. To this nd, w propos nw futur-awar onlin ski-rntal algorithms and build nw onlin solutions. W modl th availability of futur workload information as follows. For any t, th workload in th window [t, t + α ] is known, whr α [0, ] is a constant and α rprsnts th siz of th window. W prsnt both th modifid brak-vn algorithm and th dcntralizd and dtrministic onlin solution namd CSR Collctiv Srvr-Rntals) as follow. Th modifid futurawar brak-vn algorithm is vry simpl and is summarizd as th part in th srvr s actions upon ob dpartur. Futur-Awar Onlin Algorithm CSR: A cntral ob-dispatching ntity implmnts th last-mptysrvr-first ob-dispatching stratgy, i.., th on dscribd in th off-lin algorithm. Each srvr: Upon rciving a ob, th srvr starts srving th ob immdiatly. Upon a ob laving this srvr and it bcoms mpty, th srvr waits an amount of tim α). If it rcivs a ob during th priod, it starts srving th ob immdiatly. Othrwis, it looks into th look-ahad window of siz α. It turns itslf off, if it will rciv no ob during th window. Othrwis, it stays idl. In fact, as shown in Thorm 5 latr in this sction, th algorithm CSR has th bst possibl comptitiv ratio for any dtrministic algorithms. Thus, no dtrministic algorithms can achiv bttr comptitiv ratio than th algorithm CSR. Th comptitiv ratio can b improvd by rplacing th dtrministic slp dcision by a randomizd dcision, similarly to [27], but xtndd to considr futur information. Howvr, if srvrs ar turnd off aftr random tims, thn it is possibl that th last-mpty srvr is off vn though thr ar idl srvrs. Instad of using th last-mpty srvr, w will dispatch th ob to th srvr, if any, that is on but has bn idl last tim, as don in [29]. Th following dcntralizd and randomizd onlin algorithm namd RCSR Randomizd Collctiv Srvr-Rntals) is nw, and has th bst possibl comptitiv ratio. Futur-Awar Onlin Algorithm RCSR: A cntral ob-dispatching ntity implmnts th last-idl ob-dispatching stratgy. Each srvr: Upon rciving a ob: rsts all timrs, and start srving th ob immdiatly. Upon a ob laving this srvr: rcords th occupancy as a, and initializs a timr to xpir tim Z into th futur, whr Z has probability dnsity f Z z) = xpz/ { α) }) + α) α) 0<z α) + α + α δz) 7) whr δ is th Dirac dlta distribution, and X = if X is tru, and 0 othrwis. Upon xpiration of th timr, consult th prdiction ngin. If th maximum occupancy in th coming window of siz α is lss than a, thn turn off. Othrwis, rmain idl until a ob is assignd. Th following lmma, provd in Appndix C, shows that RCSR prforms at last as wll as it would undr th lastmpty-srvr-first ob-dispatching stratgy, which will allow us to obtain a comptitiv ratio. Lmma 4. For any givn workload, th cost of using RCSR is, with probability, no gratr than th cost of applying last-mpty-srvr-first with th sam pr-srvr slp policy, providd that th sam random numbr Z is gnratd undr both schms for any givn ob dpartur. Th two futur-awar onlin algorithms inhrit th nic proprtis of th proposd off-lin algorithm in th prvious sction. Th sam srvr is usd to srv a ob during its ntir soourn tim. Thus thr is no ob migration cost. Although th ob-dispatching ntity in our algorithms is cntralizd, it ust maintains a stack and stimats futur workload. Th maority of th algorithm is prformd by th dcntralizd) srvrs. This maks CSR and RCSR asy to implmnt and scalabl. Obsrving no such futur-awar onlin algorithms availabl in th litratur, w analyz thir comptitiv ratios and prsnt th rsults as follows. Assum that obs assignd to a srvr is countabl. Thorm 5. For any P, β o f f, β on, th onlin algorithms CSR and RCSR hav comptitiv ratio of 2 α and / + α). 2 α and / + α) ar th bst comptitiv ratios for dtrministic and randomizd algorithms for SCP, undr lastmpty-srvr-first ob-dispatching stratgy, rspctivly. Proof: Rfr to Appndix D. Rmarks: i) Whn α =, both two algorithms achiv th optimal srvr opration cost. This matchs th intuition

7 7 Comptitiv ratio 2.5 CSR:mpirical CR RCSR:mpirical CR CSR:analytical CR RCSR:analytical CR Look ahad window siz Figur 3: Comparison of th worst-cas comptitiv ratios according to Thorm 5) and th mpirical comptitiv ratios obsrvd in simulations using ral-world tracs. Th full-siz look-ahad window siz = 6 units of tim. Mor simulation dtails ar in Sction V. that srvrs only nd to look amount of tim ahad to mak optimal off-or-idl dcision upon ob dparturs. This immdiatly givs a fundamntal insight that futur workload information byond th critical intrval corrsponding to α = ) will not improv dynamic provisioning prformanc. ii) Th comptitiv ratios prsntd in th abov thorm ar for th worst cas. W hav carrid out simulations using ral-world tracs and found th mpirical ratios ar much bttr, as shown in Fig. 3. iii) To achiv bttr comptitiv ratios, th thorm says that it is ncssary to chang th ob-dispatching stratgy, sinc othrwis no dtrministic or randomizd algorithms do bttr than th algorithms CSR and RCSR. iv) Our analysis assums th workload information in th look-ahad window is accurat. W valuat th two onlin algorithms in simulations using ral-world tracs with prdiction rrors, and obsrv thy ar fairly robust to th rrors. Mor dtails ar providd in Sction V. Not that our algorithms ar closly rlatd to th DE- LAYEDOFF algorithm in [29], dspit th fact that thy sk to optimiz diffrnt obctiv functions total nrgy consumption in our study v.s. Enrgy-Rspons tim Product ERP) in [29]). Th main algorithmic diffrnc is that w mak us of futur information to improv prformanc, and us randomization to improv th comptitiv ratio. Th main analytic diffrnc is that w considr worst-cas prformanc, whras [29] considrs xpctd prformanc in a stochastic stting and a larg-systm asymptotic rgim. C. Adapting th Algorithms to Work with Discrt-Tim Fluid Workload Modl Adapting our off-lin and onlin algorithms to work with th discrt-tim fluid workload modl involvs two simpl modifications. Rcall in th discrt-tim fluid modl, tim is dividd into qual-lngth slots. Jobs arriving in on slot gt srvd in th sam slot. Workload can b split among running srvrs at arbitrary granularity lik a fluid. For th ob-dispatching ntity in all th algorithms, at th nd of ach slot whn all srvrs ar considrd to b mpty, it pushs all th srvr IDs back into th stack ordr dosn t mattr). Thn at th bginning of ach slot, it pops ust-nough srvr IDs from th stack in a Last-In/First-Out mannr to satisfy th currnt workload. In this way, th ob-dispatching ntity ssntially packs th workload to as fw srvrs as possibl, following th last-mpty-srvr-first stratgy. Each individual srvr starts to srv upon rciving a ob, and starts to solv th off-lin or onlin) discrt skirntal problm upon th ob laving. It is not difficult to vrify th modifid algorithms still rtain thir corrsponding prformanc guarants. Spcifically: Corollary 6. Th modifid dtrministic and randomizd onlin algorithms for discrt-tim fluid workload hav comptitiv ratios of 2 α and θ / θ + α), rspctivly, whr θ is th diffrnc btwn th lngth of brak-vn intrval and look-ahad window and θ = + /θ) θ as θ. D. Extnding to Cas Whr Srvrs Hav Stup Tim. Until now, w hav ignord th tim T s rquird for a srvr to turn on. Th comptitiv ratio will b unboundd unlss thr is a bound on th numbr of obs that can arriv during T s. W now dscrib a cntralizd algorithm EXT that provids a boundd CR in th cas whr aτ) +γ)at) for all τ [t, t + T s ]. W xpct γ to b small [30]. In this modl, srvrs can b in thr stats: ON, BOOT, OFF. Only srvrs in stat ON can srv obs, but srvrs in stats ON and BOOT both consum powr P pr unit tim. An OFF srvr turns on whn it ntrs stat BOOT; T s latr it will bcom ON. A srvr in any stat can immdiatly b turnd OFF. Algorithm EXT for Cass with Stup Tim: Each srvr: Bhavs as for CSR or RCSR, but whn its timr xpirs, it dos not turn off but snds a mssag M to managr. Managr: Kps track of th st X of siz x) of activ srvrs, i.., thos that hav not snt M sinc bing allocatd a ob. It rsponds to two typs of vnts as follows: Job arrival: If X contains an idl srvr, th ob is snt to a srvr in X using last-idl. Othrwis it is snt to anothr ON srvr. Additional srvrs will b turnd on so that th total numbr of ON and BOOT srvrs is x + γ) +. Mssag M from srvr: All but x + γ) + srvrs will b turnd OFF. BOOT srvrs ar turnd off first, in dcrasing ordr of how rcntly thy wr turnd on. No activ srvrs ar turnd off. Th following rsult, provn in Appndix E, stablishs th validity and prformanc guarants of EXT. Corollary 7. If thr ar a 0) + γ) + ON srvrs at tim 0, thn undr EXT, th numbr of ON srvrs at tim t is at last a t). Lt a min = min t [0,T] at). Th comptitiv ratio of EXT on instancs with discrt arrival instants is 2 α) + γ) + 2/a min if srvrs us CSR, or +α + γ) + 2/a min if srvrs us RCSR. Ths ar boundd abov by 2 α) + γ) + 2γ and +α + γ) + 2γ. Rmarks: i) Th comptitiv ratio of EXT is linarly proportional to γ. ii) Sinc th minimal workload a min in

8 8 larg data cntrs is normally much largr than that in small ons, whnc EXT is usually mor bnficial for larg data cntr. iii) In EXT, w adopt ovr-provisioning to combat th problm that srvrs nd stup tim T s, howvr, futur workload may not b known; it would b intrsting to know if thr xist othr approachs to handl this problm. EXT can not achiv comptitiv ratio of vn if α =. Thrfor, it is also good to know how to bttr utiliz futur information. V. Exprimnts W implmnt th proposd off-lin and onlin algorithms and carry out simulations using ral-world tracs to valuat thir prformanc. Our aims ar thrfold. First, to valuat th prformanc of th algorithms in a typical stting. Scond, to study th impacts of workload prdiction rror and workload charactristics on th algorithms prformanc. Third, to compar our algorithms to two rcntly proposd solutions LCPw) in [22] and DELAYEDOFF in [29]. A. Sttings Workload trac: Th tracs w us in xprimnts ar a st of I/O tracs takn from 6 RAID volums at MSR Cambridg [3]. Th tracd priod was on wk from Fbruary 22 to 29, W stimat th avrag numbr of obs ovr disoint 0 minut intrvals. Th data trac has a pak-to-man ratio PMR) of Th obs ar rqust-rspons typ and thus th workload is bttr dscribd by a discrt-tim fluid modl, with th slot lngth bing 0 minuts and th load in ach slot bing th avrag numbr of obs. In th xprimnts, w run algorithm LCPw) [22] by dirctly using th abov discrt-tim trac, sinc LCPw) was originally dsignd to work undr a discrt-tim stting. Manwhil, CSR, RCSR, and DELAYEDOFF [29] wr primally dsignd to work undr a continuous-tim stting. To valuat thir prformanc by using th abov discrt-tim trac, w run ths algorithms by fding obs continuously to th algorithms, whr th ob-arrivals in a slot ar assumd to uniformly sprad out th slot. By this stting, w would lik to dmonstrat that algorithms CSR/RCSR/DELAYEDOFF do not rquir to know th numbr of ob-arrivals a priori to oprat. W us last-mpty-srvr-first for RCSR. Cost bnchmark: Currnt data cntrs usually do not us dynamic provisioning. Th cost incurrd by static provisioning is usually considrd as bnchmark to valuat nw algorithms [22], [6]. Static provisioning runs a constant numbr of srvrs to srv th workload. In ordr to satisfy th timvarying dmand during a priod, data cntrs usually ovrly provision and kp mor running srvrs than what is ndd to satisfy th pak load. In our xprimnt, w assum that th data cntr has th complt workload information ahad of tim and provisions xactly to satisfy th pak load. Using such bnchmark givs us a consrvativ stimat of th cost saving from our algorithms. Svr opration cost: Th srvr opration cost is dtrmind by unit-tim nrgy cost P and on-off costs β on and β o f f. In th xprimnt, w assum that a srvr consums on unit nrgy for pr unit tim, i.., P =, x. W st β o f f + β on = 6, i.., th cost of turning a srvr off and on onc is qual to that of running it for six units of tim [22]. Undr this stting, th critical intrval is = β o f f + β on ) /P = 6 units of tim. B. Prformanc of th Proposd Onlin Algorithms W hav charactrizd in Thorm 5 th comptitiv ratios of CSR and RCSR as th look-ahad window siz, i.., α, incrass. Th rsulting comptitiv ratios, i.., 2 α and / + α), alrady appaling, ar for th worst cas. In practic, th actual prformanc can b vn bttr. In our first xprimnt, w study th prformanc of CSR and RCSR using ral-world tracs. Th rsults ar shown in Fig. 4b. Th cost rduction curvs ar obtaind by comparing th cost incurrd by th off-lin algorithm, CSR, RCSR, th LCPw) algorithm [22] and th DELAYEDOFF algorithm [29] to th cost bnchmark. Th vrtical axis indicats th cost rduction and th horizontal axis indicats th siz of lookahad window varying from 0 to 0 units of tim. For this workload, CSR, RCSR, LCPw) and DELAYED- OFF achiv substantial cost rduction as compard to th bnchmark. In particular, th cost rductions of CSR and RCSR ar byond 66% vn whn no futur workload information is availabl. LCPw) starts to prform whn th lookahad window siz is on. This is bcaus w run LCPw) undr a discrt-tim stting and th workload information for th currnt slot is only availabl aftr all obs in this slot hav arrivd. Manwhil, CSR, RCSR, and DELAYEDOFF ar running undr a continuous-tim stting, whr obs arriving at any momnt ar srvd immdiatly. Th cost rductions of CSR and RCSR grow linarly as th look-ahad window incrass, and raching optimal whn th look-ahad window siz rachs. Ths obsrvations match what Thorm 5 prdicts. Manwhil, LCPw) has not yt rach th optimal prformanc whn th look-ahad window siz rachs th critical valu. DELAYEDOFF has th sam prformanc for all look-ahad window sizs sinc it dos not xploit futur workload information. C. Impact of Prdiction Error Prvious xprimnts show that CSR, RCSR and LCPw) hav bttr prformanc if accurat futur workload is availabl. Howvr, thr ar always prdiction rrors in practic. Thrfor, it is important to valuat th prformanc of th algorithms in th prsnt of prdiction rror. To achiv this goal, w valuat CSR and RCSR with look-ahad window siz of 2 and 4 units of tim. Zro-man Gaussian prdiction rror is addd to ach unit-tim workload in th look-ahad window, with its standard dviation grows from 0 to 50% of th corrsponding actual workload. In practic, prdiction rror tnds to b small [32]; thus w ar ssntially strss-tsting th algorithms. W avrag 00 runs for ach algorithm and show th rsults in Fig. 4c, whr th vrtical axis rprsnts th cost rduction as compard to th bnchmark.

9 9 Workload Timhour) a) MSR data trac for on wk %Cost rduction 70 Opt CSR 68 RCSR LCP Dlaydoff look ahad window siz 0 b) Impact of futur information %Cost rduction Window Siz:4 68 Window Siz:2 67 Opt CSR RCSR LCP Prdiction rror%) 50 c) Impact of prdiction rror %Cost rduction 80 Opt 60 CSR RCSR LCP 40 Dlaydoff Pak to man ratio d) Impact of PMR Figur 4: Ral-world workload trac and th prformanc of algorithms undr diffrnt sttings. Th critical intrval is 6 units of tim. W discuss th prformanc of algorithms CSR, RCSR, LCPw) and DELAYEDOFF in Sction V-E. On on hand, w obsrv all algorithms ar fairly robust to prdiction rrors. On th othr hand, all algorithms achiv bttr prformanc with look-ahad window siz 4 than siz 2. This indicats mor futur workload information, vn inaccurat, is still usful in boosting th prformanc. D. Impact of Pak-to-Man Ratio PMR) Intuitivly, comparing to static provisioning, dynamic provisioning can sav mor powr whn th data cntr trac has larg PMR. Our xprimnts confirm this intuition which is also obsrvd in othr works [22], [6]. Similar to [22], w gnrat th workload from th MSR tracs by scaling a t) as a t) = Ka γ t), and adusting γ and K to kp th man constant. W run th off-lin algorithm, CSR, RCSR, LCPw) and DELAYEDOFF using workloads with diffrnt PMRs ranging from 2 to 0, with look-ahad window siz of on unit tim. Th rsults ar shown in Fig. 4d. As sn, nrgy saving incrass form about 40% at PRM=2, which is common in larg data cntrs, to larg valus for th highr PMRs that is common in small to mdium sizd data cntrs. Similar rsults ar obsrvd for diffrnt look-ahad window sizs. E. Discussion Not that CSR and RCSR hav comptitiv ratios 2 α and / + α), which improv as futur information is availabl. This is in contrast to LCPw), whos bst known comptitiv ratio is 3 and, rgardlss of how much futur information is availabl, thr ar instancs with prformanc arbitrarily clos to th ratio. Fig. 4b shows that CSR/RCSR prform slightly bttr than LCPw), partially bcaus thy nd not work in discrt tim. Ths prformanc gains of CSR/RCSR ovr LCPw) and DELAYEDOFF shown in Fig. 4b, whn multiplying th larg amount of nrgy consumd by th data cntrs vry yar, corrspond to non-ngligibl nrgy cost saving. Morovr, th slp managmnt in CSR/RCSR ar dcntralizd, which maks thm vry much asir to implmnt; whil th LCPw) is inhrntly cntralizd, sinc it rquirs th solution of a convx program at ach tim. Although in this xampl, DELAYEDOFF prforms clos to th optimal, thr ar vry natural cass in which it can b almost a factor of two mor xpnsiv than CSR/RCSR. Th valu of is approximatly on hour [22], and it is common for workloads to hav a priodic structur with priod on hour. In this cas, it is possibl that DELAYEDOFF always turns machins off ust bfor thy ar ndd again. If th workload can b prdictd an hour into th futur, thn CSR/RCSR can guarant optimal prformanc in this cas. DELAYEDOFF also dos not xploit randomnss to improv prformanc lik RCSR dos. VI. Concluding Rmarks Dynamic provisioning is an ffctiv tchniqu for rducing srvr nrgy consumption in data cntrs, by turning off unncssary srvrs to sav nrgy. In this papr, w dsign onlin dynamic provisioning algorithms with zro or partial futur workload information availabl. W rval an lgant divid-and-conqur structur of th off-lin dynamic provisioning problm, undr th cost modl that a running srvr consums a fixd amount of nrgy pr unit tim. Exploiting such structur, w show its optimal solution can b achivd by th data cntr adopting a simpl last-mpty-srvr-first ob-dispatching stratgy and ach srvr indpndntly solving a classic ski-rntal problm. W build upon this architctural insight to dsign two nw dcntralizd onlin algorithms. On is dtrministic with comptitiv ratio 2 α, whr 0 α is th fraction of th full-siz look-ahad window in which futur workload information is availabl. Th siz of th full-siz look-ahad window is dtrmind by th war-and-tar cost and th unittim nrgy cost of running a singl srvr. Th othr is randomizd with comptitiv ratio / + α). Th ratios 2 α and / + α) ar th bst comptitiv ratios for any dtrministic and randomizd onlin algorithms undr last-mpty-srvr-first ob-dispatching stratgy. Not that th problm w study in this papr is similar to that studid in [22]. Th diffrnc is that w optimiz a linar cost function ovr intgr variabls, whil Lin t al. in [22] minimiz a convx cost function ovr continuous variabls by rlaxing th intgr constraints). This papr and [22] obtain diffrnt onlin algorithms with diffrnt comptitiv ratios for th two diffrnt formulations, rspctivly. Our rsults lad to a fundamntal obsrvation that undr th cost modl that a running srvr consums a fixd amount of nrgy pr unit tim, futur workload information byond th th full-siz look-ahad window will not improv th dynamic provisioning prformanc. In addition, w also propos onlin algorithms for th cas that srvrs nd stup tim T s but th load satisfis aτ) + γ)at) for all τ [t, t + T s ]. Ths algorithms hav comptitiv ratios 2 α) + γ) + 2γ and +α + γ) + 2γ.

10 0 Our algorithms ar simpl and asy to implmnt. Simulations using ral-world tracs show that our algorithms can achiv clos-to-optimal nrgy-saving prformanc, and ar robust to futur-workload prdiction rrors. Ths rsults suggst that it is possibl to rduc srvr nrgy consumption significantly with zro or only partial futur workload information. This work can b xtndd in many important dirctions. In th lphant modl considrd hr, ach srvr could only srv on ob at a tim. Cloud data cntrs typically run multipl VMs on ach physical machin. On particular motivation is to pack togthr obs with complmntary rsourc rquirmnts, such as placing a CPU-intnsiv and a mmory-intnsiv VM on th sam srvr. In this scnario, minimizing th total powr cost is a dynamic bin-packing problm which is NP-hard. It contains classic bin packing as a spcial cas.). Th analysis of dynamic bin-packing problm is ntirly diffrnt and it would b intrsting to look at it in th futur. Evn in th simplst cas that ach srvr can host an arbitrary combination of m VMs, th problm is significantly diffrnt; it is no longr th cas that th optimal prformanc can b obtaind by a non-clairvoyant algorithm without VM migration, and indd such algorithms ar at bst m- comptitiv. A rlatd xtnsion would b to considr th fact that VMs may hav tim-varying rsourc rquirmnts. Anothr important dirction would b to xtnd ths rsults to th cas of htrognous srvrs or multipl gographically sparatd data cntrs [33], [34], [35]. It would b usful to xtnd th insight from this papr to htrognous cass. Acknowldgmnts W thank Minghong Lin for sharing th cod of his LCP algorithm, and Eno Thrska for sharing th MSR Cambridg data cntr tracs. Tan Lu and Minghua Chn s rsarch is partially supportd by a th China 973 Program Proct No. 202CB35904), th Gnral Rsarch Fund Proct No. 4209, 400, and 40) and an Ara of Excllnc Grant Proct No. AoE/E-02/08), all stablishd undr th Univrsity Grant Committ of th Hong Kong SAR, China, as wll as an Opn Proct of Shnzhn Ky Lab of Cloud Computing Tchnology and Application and two gift grants from Microsoft and Cisco. This work was also fundd by ARC grant FT Rfrncs [] T. Lu and M. Chn, Simpl and ffctiv dynamic provisioning for powr-proportional data cntrs, in Procdings of th 46th Annual Confrnc on Information Scincs and Systms CISS), 202. [2] J. G. Koomy, Growth in data cntr lctricity us 2005 to 200, Oakland, CA: Analytics Prss, 200. [3] Spain nrgy consumption. [Onlin]. Availabl: [4] L. Barroso, Th pric of prformanc, ACM Quu, vol. 3, no. 7, pp , [5] U.S. Environmntal Protction Agncy, Epa rport on srvr and data cntr nrgy fficincy, ENERGY STAR Program, [6] Z. Liu, M. Lin, A. Wirman, S. Low, and L. 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11 [33] M. Lin, Z. Liu, A. Wirman, and L. L. H. Andrw, Onlin algorithms for gographical load balancing, in Proc. Int. Grn Computing Conf., 202. [34] R. Nathui, C. Isci, and E. Gorbatov, Exploiting platform htrognity for powr fficint data cntrs, in Autonomic Computing, ICAC 07. Fourth Intrnational Confrnc on. IEEE, 2007, pp [35] T. Hath, B. Diniz, E. Carrra, W. Mira Jr, and R. Bianchini, Enrgy consrvation in htrognous srvr clustrs, in Procdings of th tnth ACM SIGPLAN symposium on Principls and practic of paralll programming. ACM, 2005, pp [36] C. Mathiu, Onlin algorithms:ski rntal. [Onlin]. Availabl: clair/talks/skirntal.pdf Appndix Th claims mad in th prvious sctions will now b provn from A to E. A. Proof of Thorm In ordr to prov thorm, w introduc thr lmmas. Th first stablishs that P on and P o f f ar wll dfind. Lmma 8. Th optimal u s and d s hav both lft and right limits. Proof: Th intrval btwn two discontinuitis in th optimal d s or optimal u s ) is at last, and so th st of discontinuitis has no accumulation points. Sinc it is picwis constant, this is sufficint for it to hav both lft and right limits at all points. Lmma 9. Lt m = max {ā τ) : τ T s, T )}. If T s T > and m < mināt s ), āt )) thn a ncssary condition for x t) to achiv optimal cost of P ā, X, Y, T s, T ) is that x t) m, t T s, T ). Proof: Lt x i t) b any optimal solution to th abov optimization problm P ā,x, Y, T s, T ) and x i t) dos not satisfy x i t) m, t T s, T ). In ordr to prov th ncssary condition, w divid x i t) into two cass. a) If x i t) m +, t T s, T ) thn lt x t) = m, t T s, T ). Thn x i t) will consum at last T T s ) P mor powr for ach xtra running srvrs than x t) during T s, T ). On th othr hand, x t) causs at most β on + β o f f mor warand-tar cost than x i t) for turning off/on ach srvr. Bcaus T T s ) >, x i t) actually costs mor than x t), which is a contradiction with that x i t) is an optimal solution. b) Othrwis, if x i t) dos not satisfy cas a), thn thr must xist tim τ in T s, T ) such that x i τ) = m. Lt x t) = min [m, x i t)], t T s, T ). Thn x t) satisfis all th constraints of P ā,x, Y, T s, T ). Morovr, x t) dos not incur mor on-off cost or oprating cost than x i t), which mans x t) is an optimal solution. Lmma 0. Lt x t) b an optimal solution to P [ā, ā 2 ), ā T + 2 ), 2, T + 2 ], whr ā t) is dfind in sction III. Thn x t) = 0, t T, T + 2 ) 2, 0), x T) = ā T) = a T) and x 0) = ā 0) = a 0). Morovr, x t), t [0, T] is an optimal solution to SCP problm. Proof: Applying lmma 9, w hav that x t) = 0, t 2, 0). Nxt, w prov x 0) = ā 0) = a 0). Assum instad that x 0) > ā 0). Lt µ = inf{t > 0 : āt) ā0)} b th first discontinuity in ā. If āµ) = ā0) thn lt ā 0) t [ 0, µ ] ˆx t) = x t) othrwis Othrwis, lt ā 0) t [ 0, µ) ˆx t) = x t) othrwis.

12 2 Sinc x t) = 0, t 2, 0), ˆx t) incurs a lowr running cost, and no highr switching cost. This contracts th assumption that x t) is an optimal solution and so x 0) ā0). Sinc x 0) ā0) for fasibility, w hav x 0) = ā 0). By th dfinition of ā t), w hav x 0) = ā 0) = a 0). Similarly, x t) = 0, t T, T + 2 ) and x T) = ā T) = a T). Sinc x t) = 0, t T, T + 2 ) 2, 0), x T) = ā T) = a T) and x 0) = ā 0) = a 0). Suppos that x t), t [0, T] is not an optimal solution to SCP. Lt x t) dnot an optimal solution to SCP. Thn w lt x t) t [0, T] ˆx t) = x t) othrwis. Thn ˆx t) incurs lss cost than x t) in [0, T] and thy hav th sam cost in th rst priods. This contracts th optimality of x t) for P [ā, ā 2 ), ā T + 2 ), 2, T + 2 ]. Thrfor x t), t [0, T] is an optimal solution to SCP. Nxt, w ar going to prov thorm. Proof: For any µ [0, T], w must hav that µ is in som intrval τ, τ ) such that ā τ) ā µ) +, ā τ ) ā µ) + and ā t) ā µ), t τ, τ ). W divid th situation in two cass. Cas I: τ τ > In this cas, according to our Optimal Solution Construction Procdur, w will st x µ) = ā µ) = a µ). On th othr hand, according to lmma 9), x t) a µ), t τ, τ ) bcaus x t) is an optimal solution to P [ā t), ā 2 ), ā T + 2 ), 2, T + 2 ]. Thrfor, w must hav x µ) = ā µ) = a µ). This mans in Cas I our Optimal Solution Construction Procdur givs an optimal solution. Cas II: τ τ In this cas, sinc τ τ, w must hav that τ, τ [0, T]. Thrfor, thr must xist two intrvals ) τ, τ ) and τ2, τ 2 which hav following proprtis: ) τ, τ ) τ, τ ), ā τ ) ā µ) +, ā τ ) ā µ) + and τ τ. Morovr, for any intrval υ, υ) ) τ, τ with ā υ ) min [ ā τ ), ā )] ) τ + and ā υ min [ ā τ ), ā τ )] +, w must hav υ υ >. 2) τ 2, τ 2) τ, τ ), ā τ2 ) min [ ā τ ), ā τ )] +, ā τ 2) min [ā τ ), ā τ )] +, τ 2 τ 2 > and a t) min [ ā τ ), ā τ )], t τ2, τ 2). In this cas, according to our Optimal Solution Construction Procdur, w will st x µ) = min [ ā τ ), ā τ )]. On th othr hand, according to lmma 9), x t) min [ ā τ ), ā τ )], t τ2, τ 2) bcaus x t) is an optimal solution to P [ā t), ā 2 ), ā T + 2 ), 2, T + 2 ]. Thn x t) min [ ā τ ), ā τ )], t τ, τ ), bcaus turning a srvr off and on latr incurs no lss cost than ust ltting it b idl during τ, τ ) sinc τ τ. Thrfor x µ) = min [ ā τ ), ā τ )]. This mans in Cas II our Optimal Solution Construction Procdur also givs an optimal solution. Thus th x t) constructd by th Optimal Solution Construction Procdur is an optimal solution to P [ā t), ā 2 ), ā T + 2 ), 2, T + 2 ], and th rsult follows by lmma 0. B. Proof of Thorm 2 First w ar going to prov that th offlin algorithm solvs P [ā t), ā 2 ), ā T + 2 ), 2, T + 2 ]. Lmma 0 thn implis th offlin algorithm also solvs SCP. First, w ar going to prov following lmma. Lmma. Undr last-mpty-srvr-first ob dispatching, if a srvr bcoms mpty at τ and it will rciv th first ob aftr τ at τ, thn ā τ ) = ā τ ) and ā t) < ā τ ), t τ, τ ). Proof: Lt S b th ID of th srvr bcoming mpty at τ. Sinc τ is th first tim that S is poppd aftr τ, th numbr of srvr IDs blow S on th stack dos not chang during τ, τ ). At any tim, th numbr of obs in th systm is qual to th numbr of srvr s IDs that ar not in th stack of th ob-dispatching ntity. Thrfor ā τ ) = ā τ ) and ā t) < ā τ ), t τ, τ ). Lmma 2. For any idl srvr at tim µ, lt τ b th most rcnt tim bfor µ that th srvr bcam idl. Thn ā τ) > ā µ). Proof: Lt τ b th first tim aftr µ that th srvr rcivs a ob. By lmma, w hav ā t) < ā τ), t τ, τ ). Thus ā τ) > ā µ). Now, w ar going to prov thorm 2. Th proof is similar to th proof of thorm. Lt x o t) Dnot th numbr of srvrs run by th offlin algorithm at t. Proof: For any µ [0, T], w must hav that µ is in som intrval τ, τ ) such that ā τ) ā µ) +, ā τ ) ā µ) + and ā t) ā µ), t τ, τ ). W divid th situation in two cass. Cas I: τ τ > In this cas, according to our Optimal Solution Construction Procdur, w will st x µ) = ā µ) = a µ). W ar going to prov that thr is no idl srvr in th systm at µ if w ar running th proposd offlin algorithm. W will divid th situation in thr sub-cass. ) τ = 2 In this sub-cas, if thr ar idl srvrs at µ, thn som idl srvrs will rciv obs at τ. According to lmma, thr must xist a tim υ in [0, τ ) such that ā ν) > ā µ), which contradicts that ā t) ā µ), t 2, τ ). 2) τ = T + 2 In this sub-cas, if thr ar idl srvrs at µ, by lmmas and 2, thr must xist a tim υ in τ, T) such that ā ν) > ā µ), which contradicts that ā t) ā µ), t τ, T + 2 ). 3) τ, τ ) [0, T] In this sub-cas, if thr ar idl srvrs at µ, by lmmas and 2, th idl srvr will rciv a ob aftr τ. Thus th idl priod for th idl srvr is largr than τ τ >, which contradicts th fact that in our offlin algorithm no srvr is longr than. Th thr sub-cass shows that in Cas I thr is no idl srvr at µ, which mans x o µ) = x µ). Thrfor, th offlin algorithm givs an optimal solution to P [ā t), ā 2 ), ā T + 2 ), 2, T + 2 ].

13 3 Cas II: τ τ In this cas, sinc τ τ, w must hav that τ, τ [0, T]. Thrfor, thr must xist two intrvals ) τ, τ ) and τ2, τ 2 which hav following proprtis: ) τ, τ ) τ, τ ), ā τ ) ā µ) +, ā τ ) ā µ) + and τ τ. Morovr, for any intrval υ, υ) ) τ, τ with ā υ ) min [ ā τ ), ā )] ) τ + and ā υ min [ ā τ ), ā τ )] +, w must hav υ υ >. 2) τ 2, τ 2) τ, τ ), ā τ2 ) min [ ā τ ), ā τ )] +, ā τ 2) min [ā τ ), ā τ )] +, τ 2 τ 2 > and a t) min [ ā τ ), ā τ )], t τ2, τ 2). In this cas, according to our Optimal Solution Construction Procdur, w will st x µ) = min [ ā τ ), ā τ )]. Similarly to Cas I, w can also divid th situation into thr sub-cass: ) τ 2 = 2. 2) τ 2 = T ) τ 2, τ 2) [0, T]. In ach sub-cas, w can adopt th approach w usd in Cas I to show that x o t) = min [ ā τ ), ā τ )], t τ2, τ ). By lmma, th offlin algorithm will not turn off a srvr during τ, τ ). Thrfor xo µ) = min [ ā τ ), ā τ )] = x µ). In th two cass, w provd that x o t) is qual to x t) constructd by Optimal Solution Construction Procdur. Thrfor, th proposd offlin algorithm solvs SCP optimally. C. Last idl vs last mpty In ordr to prov that last-idl is at last as good as lastmpty, w ar going to prov two facts: i) th numbr of physical switchs in last-idl is no mor than last-mpty, and ii) th numbr of "on" srvrs at any givn tim undr lastidl is also no mor than that of last-mpty. Lt Ls, t) b a tim-varying prmutation of srvrs such that any arrival to or dpartur from srvr s at tim t undr lastmpty arrivs to or dparts from Ls, t) undr last-idl. Not this is a random variabl dpnding on th random variabls Z chosn at tims prior to t.) Multipl such prmutations xist; w impos th continuity condition so that Ls, t) only changs at ob arrival tims. Spcifically, if Ls, t ) = Ls 2, t 2 ) for s s 2 and t < t 2 thn thr is an arrival to ithr s or s 2 undr last-mpty in th intrval [t, t 2 ], and last-idl assigns th ob to a diffrnt srvr. Partition th intrval [0, T) as follows. Lt D s b th st of points of discontinuity of Ls, ). W claim that, with probability, thr ar no accumulation points in D s. To s this, not that an accumulation point would only occur if thr wr an intrval of lngth ɛ such that thr wr an infinit numbr of i.i.d.) random timouts Z gnratd, ach of which is lss than ɛ. W can thn partition [0, T) into intrvals of th form [a s i), a s i + )), whr a s ) D s. According to th continuity condition of L s, t), all th points a s ) in D s ar ob arrival points. W can think of Ls, ) as dfining a logical srvr that srvs th sam obs undr last-idl as s dos undr lastmpty. By hypothsis, it also gnrats th sam squnc of Z slp timouts undr last-idl as s dos undr last-mpty. Thr ar two ways that logical srvr L s, ) can turn on ar: i) th mapping L s, ) rmains constant and th srvr Ls, t) turns on. ii) th mapping L s, ) changs from a srvr that is off to a srvr that is on. Consquntly, th only tims that logical srvr Ls, ) turns on or off and srvr s dos not ar in th cas ii). Ths switchs do not corrspond to a physical srvr turning on or off, and so do not incur a switching cost. Hnc th total switching cost undr last-idl is at most that undr last-mpty. It rmains to show that x Ls,t) t) undr last-idl is at most x s t) undr last-mpty. Th only caus for Ls, ) to turn on is a nw arrival, aftr which both s and Ls, t) must b on. Th only tims that s turns off that Ls, ) dos not ar during idl priods whn Ls, ) has alrady turnd off for fr du to a discontinuity in Ls, ). D. Proof of Thorm 5 In ordr to prov thorm 5, w us Lmma 3 and two othr tchnical lmmas. First, lt us introduc som notation. Lt τ,s b th tim in [0, T] that ob arrivs at srvr s, and τ, b th tim that lavs th systm. Lt τ,s = inf{t > τ, : a ob arrivs to sat tim t}; if thr ar finitly many arrivals in [0, T] thn τ,s = τ +,s. W also considr tim T as a virtual ob arrival point to th srvr. Lmma 3. Th dtrministic onlin ski-rntal algorithm usd by th onlin algorithm CSR has comptitiv ratio 2 α. Proof: As w alrady provd in Lmma 3, for both onlin and off-lin cass, a srvr facs th sam st of obs. From now on, w focus on on srvr, s. Th srvr should dcid to turn itslf off or stay idl btwn τ, and τ,s. In ordr to find th comptitiv ratio, w compar th costs P on and P o f f of th onlin ] and off-lin ski-rntal algorithms rspctivly in τ,s, τ,s. This dos not includ th cost to turn on at τ,s, but dos includ th cost to turn on at τ,s or immdiatly aftr if τ,s is an accumulation point). Th costs of th onlin and off-lin ski-rntal algorithms dpnd on th lngth of th tim btwn τ, and τ,s. Lt T,B = τ, τ,s dnot th lngth of th busy priod in τ,s, τ,s] and T,E = τ,s τ, dnot th lngth of th mpty priod in τ,s, τ,s]. Thn P o f f = P b T,B + PT,E P b T,B + ) β on + β o f f i f T,E i f T,E > and th onlin ski-rntal algorithm in CSR givs P on = 8) P b T,B + PT,E i f T,E P b T,B + ) β on + β o f f + P α) i f T,E > 9) Hnc, T,E implis P ) on βon + β o f f, T,E > implis P on P o f f /P o f f =, and sinc P = βon + β o f f ) + P α) βon + β o f f ) = 2 α. In ithr cas, P on 2 α for any T,E. Summing ovr and s givs th rsult. /P o f f

14 4 Lmma 4. Th randomizd onlin ski-rntal algorithm usd by th onlin algorithm RCSR with last-mpty-srvr-first stratgy has comptitiv ratio / + α). Proof: In th proof, w again focus on on srvr. W will us th notation usd to prov Lmma 3. This tim it is sufficint to compar th avrag cost P on of th randomizd onlin ski-rntal algorithm in τ,s, τ,s] with off-lin optimal cost in 8). Undr th randomizd onlin ski-rntal algorithm, whn T < α, w hav E ) P on = Pb T,B + PT,E ; whn α T,E, w hav E ) P on = P b T,B + +P α) T,E α and whn T,E >, w hav E ) P on = Pb T,B + T,E α 0 Pz + βon + β o f f ) fz z) dz T,E f Z z) dz; α) 0 Pz + βon + β o f f ) fz z) dz. W gt th abov xpctd cost for α T,E as follows: If th numbr Z gnratd by th srvr is lss than T,E α, thn th srvr will wait for tim Z, consuming nrgy PZ. It looks into th look-ahad window of siz α and finds it won t rciv any ob during th window bcaus Z < ) T,E α. Thrfor, it turns itslf off and costs βon + β o f f. On th othr hand, if Z T,E α, th srvr will not turn itslf off and consum PT,E to stay idl. W can gt th xpctd cost for T,E < α and T,E > in th sam way. According to th distribution of Z in RCSR, w can calculat E ) ) P,on and th ratio btwn E P,on and P,o f f : E ) P on, T,E < α = T,E α P o f f +α From this xprssion, for all, w can conclud that E ) P on /P o f f +α for any T,E. Summing ovr and s givs th rsult. Now w ar rady to prov thorm 5. Rcall that th optimal cost of th data cntr can b achivd by ach srvr running an off-lin ski-rntal algorithm indpndntly. On th othr hand, in Lmmas 3 and 4, w provd that th cost of dtrministic and randomizd onlin ski-rntal algorithm w applid ar at most 2 α and +α tims th cost of off-lin ski-rntal algorithm for on srvr. Thrfor, th cost of our onlin algorithm CSR is at most 2 α tims th cost of off-lin algorithm for data cntr. Morovr, if w adopt last-mpty-srvr-first ob-dispatching stratgy in th randomizd algorithm RCSR, it can achiv comptitiv ratio +α. By Lmma 4, RCSR with last-idl prforms at last as wll as if it adoptd last-mpty-srvr-first ob-dispatching stratgy. Thrfor, RCSR has comptitiv ratio +α. Nxt, w want to prov that CSR has th bst comptitiv ratio for dtrministic onlin algorithms. First, w prov that th bst comptitiv ratio of a dtrministic algorithm for a singl ski-rntal problm is 2 α. Assum that th dtrministic onlin algorithm pks into th look-ahad window and thn dcid to turn off or stay idl tim θ aftr bcoming mpty at t. For θ < α, if th srvr rcivs its nxt ob right aftr t + θ + α), thn th onlin algorithm will turn off itslf at t + θ, and consum nrgy P θ + ). On th othr hand, th offlin optimal is α + θ) P, whnc th comptitiv ratio is at last θ+ θ+α > 2 α. For θ > α, if th srvr rcivs its nxt ob right aftr t + θ + α), thn th onlin algorithm will turn off itslf at t + θ, and consum P θ + ) powr. On th othr hand, th offlin optimal is P. Th comptitiv ratio at last is + θ > 2 α. Hnc, only whn θ = α can th dtrministic algorithm hav th comptitiv ratio 2 α. Thrfor, th bst comptitiv ratio of dtrministic algorithm for a singl ski-rntal problm is 2 α. Howvr, a srvr will rciv a squnc of obs in data cntr. And aftr finishing ach ob, th srvr facs a ski-rntal problm. Hnc, ach srvr actually facs a rpatd ski-rntal problm. As for rpatd ski-rntal problm w hav following lmma. Lmma 5. Th bst comptitiv ratio of a dtrministic algorithm for th rpatd ski-rntal problm facd by ach srvr is 2 α. Proof: W will prov lmma 5 by induction. Assum that dtrministic algorithm A achivs th bst comptitiv ratio for rpatd ski-rntal problm. Lt θ i b th lngth of idl tim bfor th srvr pks into th look-ahad in th ith ski-rntal problm. Sinc th bst dtrministic algorithm for singl ski-rntal problm must pk into th look-ahad window aftr staying idl for α), A must hav θ = α. Suppos θ i = α for i =, 2,...k. Thrfor, th cost of A is up to 2 α tims th off-lin optimal for th first k ski-rntal problms. W will prov that A must hav θ k+ = α. If θ k+ < α or θ +k > α, w can us th sam approach w usd to prov th bst comptitiv ratio for singl ski-rntal is 2 α to show that in th k + )th ski-rntal problm A consums mor than 2 α tims th off-lin optimal in worst cas for this phas. If th worst cas of this phas occurs aftr th worst cas of th prvious phass, this would rsult in an ovrall comptitiv ratio xcding 2 α. Thrfor, w must hav θ k+ = α if th comptitiv ratio is to b at most 2 α. Sinc th schm with θ k = α for all k has comptitiv ratio xactly 2 α, it follows that th bst comptitiv ratio of dtrministic algorithm for rpatd ski-rntal algorithm is 2 α. Whn a t), th problm SCP w study in this papr bcoms a rpatd ski-rntal problm. Thrfor, th bst comptitiv ratio for a dtrministic onlin algorithm is 2 α, which is achivd by CSR. Finally, w want to prov that RCSR has th bst comptitiv ratio for randomizd onlin algorithms. Considr th cas that th srvr bcoms mpty at τ and it will rciv its nxt ob at τ 2. In ordr to find th bst comptitiv ratio for a randomizd onlin algorithm, according to th proof of Lmma 4, it is sufficint to find th minimal ratio of th cost by a randomizd onlin algorithm to that of th offlin

15 5 optimal in [τ, τ 2 ]. Th comptitiv ratio cannot b lowr than th comptitiv ratio on an instanc with a singl mpty intrval, and so w considr that cas. W first divid tim priod τ, τ 2 ) into slots of qual lngth. As th lngth of th slots gos to zro, w can gt th bst comptitiv ratio for a continuous tim randomizd onlin algorithm. Assum th critical intrval contains xact b slots and thr ar D slots in [τ, τ 2 ]. W focus on th cas that th look-ahad window has k b 2 slots. If k b, th onlin algorithm can achiv th offlin optimum and th comptitiv ratio is.) Lt p i dnot th probability that th algorithm dcids to turn off th srvr at slot i =, 2,.... Lt th comptitiv ratio b c. Rgardlss of th valu of D, th xpctd onlin cost must b at most th comptitiv ratio tims th off-lin cost. Thus th minimum comptitiv ratio satisfis inf c 0) s.t. D p i cd, D [0, k], ) D k b + i ) p i + D k b + i ) p i + i=d k+ i=d k+ Dp i Dc, D k, b]2) Dp i bc, D b, ]3) p i =, 0 p i, i 4) var c, p i, i {, 2, 3,...} 5) W can apply th stps in [36] to show that th optimal valu c d of problm 0) 5) is qual to th optimal valu c of following problm. min c 6) s.t. c, D [0, k], 7) D k b k b + i ) p i + D p i D c, D k, b)8) b k i=d k+ b + i ) p i b c, D [b, ] 9) b k p i = 0 p i, i 20) var c, p i, i {, 2,..., b k} 2) Nxt, w prov that p is positiv. If instad p = 0, lt b th minimal i such that p i > 0. Thn th constraints 8) 9) must hold as strict inqualitis for D k +, for th following rason. First considr th constraint for D = k +. Sinc w hav b k othrwis w obtain th dtrministic algorithm CSR, which is suboptimal), w hav D = k + < b and th constraint for D = k +, dividd by D, is b + b k p + p i c k + i= + and whn D k +, th constraints, dividd by D, ar b k p i c. i= Sinc k b 2, if th lattr wr activ, thn th formr would b violatd. W us th slacknss of ths constraints to show p > 0. Th cofficint of p is lss than that of p in th constraints for D > k+. Thrfor, w can dcras p a littl bit and incras p a littl bit such that all th constraints of 7) 20) hav slacknss, which mans w can find a smallr c which satisfis all th constraints. This contradicts th optimality of p = [ p, p 2, p 3,, b k] p. Thrfor, w must hav p > 0. Nxt, w again follow [36] to show that ach of th inqualitis in 8), 9) is tight. Assum instad that th constraint corrsponding to som particular D k, b] is loos. Lt D # b th largst such D. Considr cas i) that D # < b. Not that p D # k+ > 0, sinc othrwis D# + would also b slack. Thn dcras p and incras D # k+ p slightly. This dos not affct constraints for smallr D # k D, but introducs slack into th constraints for all largr D. Nxt, w could incras p D # k and dcras p, which dosn t affct constraints for largr D, but introducs slack for all constraints with smallr D. Altrnativly, in cas ii) that D # = b, w can dcras p whil incrasing p b k to introduc slack into arlir constraints. In ithr cas, th transformation inducs slack in all constraints, which allows c to dcras, contradicting th optimality of c. Thrfor, all th constraints for D k, b] must b tight. Sinc th total b k constraints for all th D k, b] is tight and b k p i =, w can solv th systm of linar quations and gt th minimal comptitiv ratio and probability distribution: ) b k b k c = b k b k b ) i p b k i = c b k, 0 i < b k b k b k ) b k b k p = k + b k b c, k < b Ltting b go to infinity and kping k/b = α, w hav th minimal comptitiv ratio c for continuous tim: c = + α This mans th optimal comptitiv ratio for continuous tim randomizd onlin algorithm is c = +α, as rquird. Thrfor, th bst comptitiv ratio of randomizd algorithm for singl ski-rntal problm is +α. W hav th following lmma to prov that RCSR has th bst comptitiv ratio for randomizd algorithms against oblivious advrsary. Lmma 6. Th bst comptitiv ratio of randomizd algorithm for th rpatd ski-rntal problm is +α. Proof: In th proof w will us th notation usd in th proof of lmma 3. Assum that th cost of onlin algorithm

16 6 in th ith ski rntal is C i and th corrsponding offlin optimal is Ci. If th stratgy of th oblivious advrsary is to arbitrarily choos a numbr which is gratr than α as th mpty priod of ach ski rntal problm, thn th onlin algorithm has no information of th lngth of currnt mpty priod T i,e vn if th onlin algorithm knows T i,e, T i 2,E,...T,E. W ar going to prov lmma 6 by induction. It is clar that in th first ski rntal w can not do bttr than in singl ski rntal problm. Thrfor, w hav E C ) +α C. Assum that E ) k k E C i +α Ci. ) k C i k +α Ci, w ar going to prov Suppos onlin algorithm chooss f Zk z k z k, z k 2...z ) as th conditional probability distribution of Z k givn th historical information. Thn thr always xists a T k,e such that E C k ) = E E C k Z k, Z k 2...Z )) + α C k To s this, suppos thr is no T k,e satisfying abov inquality, thn for any T k,e, w must hav E E C k Z k, Z k 2...Z )) < + α C k Thn in th singl ski rntal problm, w can also lt th distribution of random variabl Z follow th unconditional distribution f Zk z k ) of Z k. In this way, w can gt a bttr comptitiv ratio than +α for singl ski rntal problm. This is a contradiction. Thrfor, such T k,e must xist. This mans th ) bst ratio ) w can do in kth ski rntal is +α. Sinc k k E C i = E C i + E C k ) and k Ci = k Ci + C k, thus w hav k E C i k Ci + α as rquird. This mans onlin algorithms can not do bttr than +α vn against oblivious advrsary. Thrfor, RCSR has th bst comptitiv ratio +α for randomizd algorithms against oblivious advrsary. Sinc SCP has rpatd ski rntal problm as a spcial cas, +α is th optimal comptitiv ratio for any randomizd algorithm. E. Proof of Corollary 7 In this sction, w ar going to prov corollary 7. Proof: W prov th rsult for RCSR. Sinc w mak no us of th form of f X, th sam proof holds for CSR which corrsponds to RCSR with f x x) = δx )). W first stablish validity. Not that xt) is th numbr of srvrs ON undr RCSR, and that x incrass by at most a factor of + γ in an intrval of lngth T s, sinc xt) at) at th start, and xt) = at) at all tims that x incrass. Sinc arrival instants ar discrt, thr ar also no limit point in th N st of tims t n n=0 at which x changs, and so w can apply induction on n. By induction, th numbr of ON and BOOT srvrs at ach tim t n is ithr xt n ) + γ) or xt n ) + γ) +. Th bas cas, t 0 = 0, is tru by hypothsis. For subsqunt t n it is tru by construction xcpt that whn M is snt, thr may b only xt n ) + γ) + xt n ) + γ) srvrs ON or BOOTing. W now show by induction that thr ar at last xt n ) ON srvrs at ach tim t n. If x dcrass at t n, this is tru sinc thr wr at last xt n ) > xt n ) srvrs ON bfor t n. Nxt considr th cas that x incrass at t n. Lt τ = arg min τ [tn T s,t n ] xτ), with tis brokn by taking th smallst τ. W claim all BOOT srvrs at τ wr BOOT at t n T s. This is trivial if τ = t n T s. To s it in othr cass, suppos instad thr is a BOOT srvr at τ that was turnd on at τ t n T s, τ). Now xτ ) > xτ) by th minimality of τ, and so xτ ) + γ) xτ) + γ) +, whnc thr ar mor ON or BOOT srvrs at τ than at τ. Howvr, sinc EXT turns of th most rcntly turnd on BOOT srvrs first, xistnc at τ of a BOOT srvr turnd on at τ mans that mor srvrs ar turnd on during [τ, τ] than ar turnd off, which is a contradiction. Sinc x t n ) x τ) + γ) and all th BOOT srvrs at τ will bcom ON at t n, thus thr will b at last x τ)+ x τ) γ% + x t n ) ON srvrs. This complts th induction. Sinc x t) a t), w hav provd th numbr of ON srvrs is at last a t) at tim t in th xtndd algorithm, which stablishs th first claim of th thorm. To prov th comptitiv ratios, not that th numbr of total activ srvrs in EXT is at most + γ) x t) + 2. Th total running nrgy cost of EXT is at most 2PT mor than + γ) tims th running cost of RCSR. Now, w ar going to analyz th switching cost. W divid th x t) down into priods during which x t) is incrasing and priods in which it is dcrasing. Morovr, a dcrasing/incrasing priod must b followd by a incrasing/dcrasing priod and th combination of all th priods covrs th intrval of [0, T]. In any incrasing priod, assum that x t) incras from A to A + k, th numbr of turning-on in xtndd algorithm is A + k) + γ) A + γ) k + γ) W will gt similar rsult for dcrasing priod. Thrfor, th total switching cost of th xtndd algorithm is at most + γ) tims that of RCSR. Whn srvrs hav stup tim, th offlin optimal cost P S is changd. Howvr, th optimal valu of SCP is a lowr bound of P S. Hnc, th total cost of RCSR is at most +α P S. Morovr, th total cost of EXT is at most 2PT + +α + γ) P S. Th comptitiv ratio of EXT follows from that a min is th minimal workload. F. Exprimnt of Elphant Workload In this sction, w will valuat CSR and RCSR with lphant workload dfind in Sction II-A. Sinc w do not hav any ral data cntr trac of this kind of workload, w usd computr to gnrat a synthtic workload shown in Fig. 5. In this lphant workload trac, th ob arrival rat is

17 7 Poisson procss and th rat varis from an hour to anothr. Th srvic tim of ach ob is xponntially distributd and th man is about 33 hour. Sinc w nd us computr to do simulation, w chop th total tim in to small slots with lngth of 6 sconds. Th PMR of this trac is.7. Th simulation rsult is shown in Fig. 6. On obsrvation from Fig. 6 is that th CSR and RCSR nrgy-saving curvs of lphant workload ar similar to that of mic workload. Our algorithms can sav mor than 37% nrgy. And this numbr is consistnt with th rsult in Fig. 4d which indicats that th nrgy saving is about 40% whn PMR is Workload Timhour) Figur 5: Synthtic Elphant Workload. %Cost rduction Opt CSR RCSR Look ahad window siz Figur 6: Enrgy saving of our algorithms for lphant workload.