Allocations for Heterogenous Distributed Storage

Transcription

1 Allocatos fo Heteogeous Dstbuted Stoage Vasleos Ntaos Guseppe Cae Alexados G Dmaks dmaks@uscedu axv:0596v [csi] 8 Feb 0 Abstact We study the poblem of stog a data object a set of data odes that fal depedetly wth gve pobabltes Ou poblem s a atual geealzato of a homogeous stoage allocato poblem whee all the odes had the same elablty ad s atually motvated fo pee-to-pee ad cloud stoage systems wth dffeet types of odes Assumg optmal easue codg MDS), the goal s to fd a stoage allocato e, how much to stoe each ode) to maxmze the pobablty of successful ecovey hs poblem tus out to be a challegg combatoal optmzato poblem I ths wok we toduce a appoxmato famewok based o lage devato equaltes ad covex optmzato We popose two appoxmato algothms ad study the asymptotc pefomace of the esultg allocatos SUBMIED O ISI 0 I INRODUCION We ae teested heteogeous stoage systems whee stoage odes have dffeet elablty paametes hs poblem s elevat fo heteogeous pee-to-pee stoage etwoks ad cloud stoage systems that use multple types of stoage devces, eg sold state dves alog wth stadad had dsks We model ths poblem by cosdeg stoage odes ad a data collecto that accesses a adom subset of them he pobablty dstbuto of,, models adom ode falues ad we assume that ode fals depedetly wth pobablty p he pobablty of a set of odes beg accessed s theefoe: P) = p p j ) ) Assume ow that we have a sgle data fle of ut sze that we wsh to code ad stoe ove these odes to maxmze the pobablty of ecovey afte a adom set of odes fal he poblem becomes tval f we do ot put a costat o the maxmum sze of coded data ad hece, we wll wok wth a maxmum stoage budget of sze < : If x s the amout of coded data stoed ode, the x We futhe assume that ou fle s optmally coded, the sese that successful ecovey occus wheeve the total amout of data accessed by the data collecto s at least the sze of the ogal fle hs s possble pactce whe we use Maxmum Dstace Sepaable MDS) codes [] he pobablty of successful ecovey fo a allocato x,, x ) ca be wtte as j / [ ] P s = P x =,, P) x whee s the dcato fucto S = f the statemet S s tue ad zeo othewse A moe cocete way to see ths poblem s by toducg a Y Beoullp ) adom vaable fo each stoage ode: Y = whe ode s accessed by the data collecto ad Y = 0 whe ode has faled Defe the adom vaable Z = x Y ) whee x s the amout of data stoed ode he, obvously, we have P s = P[Z ] Ou goal s to fd a stoage allocato x,, x ), that maxmzes the pobablty of successful ecovey, o equvaletly, mmzes the pobablty of falue, P[Z < ] II OPIMIZAION PROBLEM Put optmzato fom, we would lke to fd a soluto to the followg poblem Q : mmze P) x x <,, x x 0, =,, Authos [] cosde a specal case of poblem Q whch p = p, Eve ths symmetc case the poblem appeas to be vey dffcult to solve due to ts o-covex ad combatoal atue I fact, eve fo a gve allocato x ad paamete p, computg the objectve fucto s computatoally tactable #P -had, See []) A vey teestg obsevato about ths poblem follows dectly fom Makov s Iequalty: P[Z ] E[Z] = p If p <, the the pobablty of successful ecovey s bouded away fom hs has motvated the defto of a ego of paametes fo whch hgh pobablty of ecovey s possble: R HP = p, ) : p he budget should be moe tha /p f we wat to am fo hgh elablty ad the authos [] showed that the above ego of paametes, maxmally speadg the budget to all odes e, x = /, ) s a asymptotcally optmal allocato as I the geeal case, whe the ode access pobabltes, p, ae ot equal, oe could follow smla steps to chaacteze

2 a ego of hgh pobablty of ecovey Makov s Iequalty yelds: P[Z ] E[Z] = x p = p x whee p = [p, p,, p ] ad x = [x, x,, x ] If we do t wat P[Z ] to be bouded away fom we have to eque ow that p x We see that ths case, hgh elablty s ot a matte of suffcet budget, as t depeds o the allocato x tself Let Sp, ) = x R + : p x, x be the set of all allocatos x wth a gve budget costat that satsfy p x fo a gve p We call these allocatos elable fo a system wth paametes p,, the sese that the esultg pobablty of successful ecovey s ot bouded away fom he the ego of hgh pobablty of ecovey ca be defed as the ego of paametes p,, such that the set Sp, ) s o-empty R HP = p, ) R + + : Sp, ) hs geealzes the ego descbed [] If all p s ae equal the the set Sp, ) s o-empty whe p x = p I the geeal case, the mmum budget such that Sp, ) s o-empty s = /p max, wth p max = maxp, ad Sp, /p max ) cotas oly oe allocato x p : x j = max p max, j = ag max p, x = 0, j Eve though R HP povdes a lowe boud o the mmum budget equed to allocate fo hgh elablty, t does t povde ay sghts o how to desg allocatos that acheve hgh pobablty of ecovey a dstbuted stoage system hs motvates us to move oe step futhe ad defe a ego of ɛ-optmal allocatos the ext secto III HE REGION OF ɛ-opimal ALLOCAIONS We say that a allocato x, x,, x ) s ɛ-optmal f the coespodg pobablty of successful ecovey, P[Z ], s geate tha ɛ Let E p,, ɛ) = x R + : P[Z < ] ɛ, x be the set of all ɛ-optmal allocatos Note that f we could effcetly chaacteze ths set fo all poblem paametes, we would be able to solve poblem Q exactly: Fd the smallest ɛ such that E p,, ɛ) s o-empty I ths secto we wll deve a suffcet codto fo a allocato to be ɛ-optmal ad povde a effcet chaactezato fo a ego H E p,, ɛ) We beg wth a vey useful lemma Lemma Hoeffdg s Iequalty [], [3]) Cosde the adom vaable W = V, whee V ae depedet almost suely bouded adom vaables wth P V [a, b ]) = he, [ ] δ P W E[W ] δ exp b a ) fo ay δ > 0 We ca use Lemma to uppe boud the pobablty of falue, P[Z < ] P[Z ], fo a abtay allocato, sce Z = x Y ca be see as the sum of depedet almost suely bouded adom vaables V = x Y, wth P V [0, x ] ) = Let δ = x p ) / ad eque δ > 0 x p > Lemma yelds: P[Z < ] exp x p ), x x p > 3) Notce that the costat x p > eques the allocato x, x,, x ) to be elable ad Sp, ) I vew of the above, a suffcet codto fo a stctly elable allocato to be ɛ-optmal s the followg exp x p ) x ɛ l /ɛ x p x, p x > We say that all allocatos satsfyg the above equato ae Hoeffdg ɛ-optmal, due to the use of Hoeffdg s Iequalty Lemma Defto he Rego of Hoeffdg ɛ-optmal allocatos H p,, ɛ) = x R + : p x >, x, x l /ɛ p x 4) 5) he above ego s stctly smalle E p,, ɛ) fo ay fte, because the boud 3) s ot geeally tght Howeve, H p,, ɛ) s a covex set: Equato 4) ca be see as a secod ode coe costat o the allocato x R + heoem he ego of Hoeffdg ɛ-optmal allocatos H p,, ɛ) s covex x hs teestg esult allows us to fomulate ad effcetly solve optmzato poblems ove H p,, ɛ) Fdg the smallest ɛ such that H p,, ɛ) s o-empty wll poduce a ɛ -optmal soluto to poblem Q A Hoeffdg Appoxmato of Q If we fx p,, as the poblem paametes, the the followg optmzato poblem ca be solved effcetly, to ay desed accuacy /α, by solvg a sequece of Olog α) covex feasblty poblems bsecto o ɛ) H : m x,ɛ ɛ st: x H p,, ɛ) We wll see ext that f s suffcetly lage, ɛ goes to zeo expoetally fast as gows, ad hece the soluto to the afoemetoed poblem s asymptotcally optmal

3 B Maxmal Speadg Allocatos ad the Asymptotc Optmalty of H Fst, we wll focus o maxmal speadg allocatos, x x R : x = /, ad deve the asymptotc optmalty fo Q, the sese that P[Z < ] 0, as Let p = p be the aveage access pobablty acoss all odes We have the followg lemma Lemma If > / p, fo ay ɛ > 0, ɛ : x H p,, ɛ), fo all ɛ Poof: hs follows dectly fom the defto of l /ɛ H p,, ɛ): ɛ = p/ ) he above lemma establshes the asymptotc optmalty of maxmal speadg allocatos though the followg coollay Coollay he pobablty of faled ecovey, P e P[Z < ], fo a maxmal speadg allocato s P e e p/ ) Whe > / p, P e 0, as he fact that H p,, ɛ) cotas maxmal speadg allocatos fo > / p, povdes a suffcet codto o the asymptotc optmalty of H heoem Let ɛ be the optmal value of H If > / p, the ɛ = Oexp)) Poof: Let > / p ad cosde the maxmal speadg allocato x he, ɛ ɛ s, whee ɛ s s the mmum ɛ such that x H p,, ɛ) hat s ɛ s = e p/ ), ad sce > / p, ɛ ɛ s = Oexp)) IV CHERNOFF RELAXAION I ths secto we take a dffeet appoach to obta a tactable covex elaxato fo Q by mmzg a appopate Cheoff uppe boud A Uppe Boudg the Objectve Fucto Lemma 3 Uppe Boud) Let Z = x Y, x 0, Y beoullp ) ad t 0 he pobablty of faled ecovey, P[Z < ], s uppe bouded by P[Z < ] g t x) = ) P) exp t x,, Poof: Fo ay t 0 we have: P[Z < ] P[Z ] = P [ e tz e t] e t E [ [ ] e tz] = e t E e txy = e t E [ e txy] = e t p + p e tx) 6) = e t =,,,, g t x) ) P) exp t x ) P) exp t x Note that g t x) s a weghted sum of covex fuctos wth lea agumets, ad hece covex x Equato 7) makes the covex elaxato of the objectve fucto appaet: x < α e txα), fo ay t 0 B he Relaxed Optmzato Poblem Befoe we move fowad ad state the elaxed optmzato poblem, we take a close look at the costat set S = x R + : x of the ogal poblem Q Fom a pactcal pespectve, t should be wasteful to allocate moe tha oe ut of data flesze) o a sgle ode If the ode suvves, the the data collecto ca always ecove the fle usg oly oe ut of data ad hece ay addtoal stoage does ot help Also, a allocato usg less tha the avalable budget caot have lage pobablty of successful ecovey I the followg lemma, we show that t s suffcet to cosde allocatos wth x [0, ] ad x = Lemma 4 Fo ay x S, x S = x R + : x =, x, =,, such that P [ x Y < ] P [ x Y < ] Poof: See the log veso of ths pape [4] he elaxed optmzato poblem ca be fomulated as follows R : mmze x g t x) x = x [0, ], =,, Note that, geeal, oe would lke to mmze f t 0 g t x) stead of g t x) fo some t 0 Howeve, fo ow, we wll let t be a fee paamete ad cay o wth the optmzato he mpotat dawback of the above fomulato hdes the objectve fucto: Although covex, g t x) has a expoetally log descpto the umbe of stoage odes: he sum s stll ove all subsets,, hs ca be ccumveted f we cosde mmzg log g t x) stead of g t x) ove the same set Lemma 5 log g t x) s covex x Poof: See the log veso of ths pape [4] 7)

4 Lemma 6 Fo ay t 0 ag m g tx) = ag m x S x S whee S = x R + : x, x log + p ) e tx, p Poof: Let x = ag m x S g t x) he g t x ) g t x), x S akg the logathm o both sdes peseves the equalty sce log ) s stctly ceasg Hece, log g t x ) log g t x), x S ad subtactg t + log p ) fom both sdes yelds the desed esult ad completes the poof I vew of Lemmas 5 ad 6, we ca solve R though the followg equvalet optmzato poblem R : mmze x t + x = log + p ) e tx p x [0, ], =,, R s a covex sepaable optmzato poblem wth polyomal sze descpto ad tems of complexty, t s ot much hade tha lea pogammg [5] Oe ca solve such poblems umecally a vey effcet way usg stadad, off-the-shelf algothms ad optmzato packages such as CVX [6], [7] C Isghts fom Optmalty Codtos fo R Hee, we move oe step futhe ad take the KK codtos fo R ode to take a close look at the stuctue of the optmal solutos Let p p he Lagaga fo R s: Lx, u, v, λ) = log ) + e tx) + λ x u x + v x ) whee λ R, u, v R + ae the coespodg Lagage multples he gadet s gve by x Lx, u, v, λ) = t + e +λu +v, ad the KK ecessay ad suffcet tx codtos fo optmalty yeld: t + λ u + v = 0, 8) + e tx x = 9) 0 x, 0) λ R, v, u 0, ) v x ) = 0, u x = 0, ) Usg the esults fom [8], the optmal soluto to R s gve by 0 f t x + λ = f λ t e t + t log ) 3) t λ f t e t + < λ < t + whee λ s chose such that Eq9) s satsfed, e, ) t log t λ λ t e t, + + λ t ) + t e t + = 4) Numecally, λ ca be computed va a teatve O ) algothm descbed [8], ad hece ths appoach gves a eve moe effcet way to solve R Howeve, the most mpotat aspect of the above esult s that we ca use equatos 3), 4) to obta closed fom solutos fo a ceta ego of poblem paametes ad aalyze the pefomace of the esultg allocatos D he choce of paamete t 0 It s clea that the optmal soluto to R depeds o ou choce of t 0 Fo example, t e t + 0, t +, as t ad x = lm t log t/λ ), Equato t 4) yelds x =, ad hece the maxmal speadg allocato becomes optmal fo R as t Eve though ths motvates the choce of maxmal speadg allocatos as appoxmate oe-shot solutos fo the ogal poblem Q, explctly tug the paamete t ca povde sgfcatly bette appoxmatos I ode to obta the tghtest boud fom Lemma 3, we have to jotly mmze the objectve R wth espect to t 0 ad x owads ths ed, oe ca teatvely optmze R by fxg the value of oe vaable t o x) at each step ad mmzg ove the othe Afte each teato the objectve fucto deceases ad hece the above pocedue coveges to a possbly local) mmum he above algothm teatvely tues the Cheoff boud toduced ths secto ad poduces a mmzg allocato that ca seve as a appoxmate soluto to the ogal poblem Q Fo aalytc puposes though, we ca choose a value fo t as follows Recall fom Lemma 3 that P[Z < ] g t x) fo ay t 0 Afte takg logathms, we would lke to fd a value fo t 0 that mmzes bt) t + log + e tx ) Notce that bt) s a covex fucto of t, wth bt) > 0, t 0, b0) = log + ) ad lm t bt) = he slope of bt) at zeo s b 0) = x + = p x, whch s egatve f the allocato s elable Whe t s lage, log + e tx ) 0, wheeas fo small values of t, log+ e tx ) tx +log ad hece bt) t + maxtx + log, 0 t + max tx + log, 0 Oe way to choose t that does ot deped o x s to make tx + log = 0 t = log

5 E A closed-fom allocato: ˆx I vew of the above esults we povde hee a closed fom allocato each x s gve as a fucto of p ad ) that ca be used to study the asymptotc pefomace of R ad seve as a bette oe-shot appoxmate soluto to Q Let E ) be a shothad otato fo the sample aveage such that Efx) = fx ), ode to smplfy the expessos Fo the above choce of t = log = Elog /, equato 3) becomes: x = 0 f f λ ) Elog log Elog λ Elog + λ Elog / e Elog / + othewse Elog / e Elog / +, 5) Lemma 7 If p > Elog, ad < log max, max = max, the x = Elog log, ) Poof: Assume that λ Elog + he fom Eq4), λ = Elog ad x = Elog log λ s deed the equed teval f Elog < >, p > /, ad Elog < e Elog /, < Elog log max Clealy, whe all p > /, ˆx : x = <, Elog / e Elog / +, Elog + Elog log,, s a feasble suboptmal allocato fo Q It s also suboptmal fo R geeal, sce solvg R va the poposed algothms ca oly acheve a smalle pobablty of faled ecovey We have P e Q P e R P Elog log Y < I the followg lemma we gve a uppe boud o the pobablty of faled ecovey fo ˆx ad establsh ts asymptotc optmalty Lemma 8 If p > Elog, ad > Ep log, the allocato ˆx : x = Elog log,, s stctly elable, ad the pobablty of faled ecovey, P e = P[Z < ], s uppe bouded by ) Ep log Elog P e exp Elog ad hece, whe > Elog Ep log, P e 0, as Poof: he poof follows dectly fom Lemma ad Equato 3) Notce that ˆx s elable fo values of fo whch a maxmal speadg allocato x Elog s ot, sce p Ep log, ad hece ts pobablty of faled ecovey P e goes to zeo expoetally fast fo smalle values of V NUMERICAL EXPERIMENS I ths secto we evaluate the pefomace of the poposed appoxmate dstbuted stoage allocatos tems of the pobablty of faled ecovey ad plot the coespodg Pobablty of faled ecovey Pe) Appoxmate DSA flesze =, N=00) 0 5 MaxSpead Pe) Cheoff CF Pe) Hoeffdg Pe) Cheoff I Pe) 0 6 MaxSpead Boud) Cheoff CF Boud) Hoeffdg Boud) Cheoff I Boud) Maxmum Avalable Stoage Budget ) Fg Pefomace of the poposed appoxmate dstbuted stoage allocatos ad the coespodg uppe bouds fo a system wth = 00 odes ad p U05, ) bouds I ou smulatos we cosde a esemble of dstbuted stoage systems wth = 00 odes, whch the coespodg access pobabltes, p U05, ), ae daw ufomly at adom fom the teval 05, ) We cosde the followg allocatos ) Maxmal speadg: x =, ) Cheoff closed-fom: x = /Elog ) log, 3) Hoeffdg ɛ-optmal: obtaed by solvg H 4) Cheoff teatve: obtaed by solvg R ad teatvely tug the paamete t Fg shows, sold les, the esemble aveage pobablty of faled ecovey of each allocato, P [ x Y < ], vesus the maxmum avalable budget I dashed les, Fg plots the coespodg bouds o P e obtaed fom Coollay, Lemma 8 ad the objectve fuctos of H, R REFERENCES [] D Leog, A Dmaks, ad Ho Dstbuted stoage allocatos CoRR, abs/0587, 00 [] W Hoeffdg Pobablty equaltes fo sums of bouded adom vaables Joual of the Ameca Stat Assocato, 5830):3 30, Mach 963 [3] M Mtzemache ad E Upfal Pobablty ad Computg: Radomzed Algothms ad Pobablstc Aalyss Cambdge Uvesty Pess, New Yok, NY, USA, 005 [4] V Ntaos, G Cae, ad A Dmaks Allocatos fo heteogeous dstbuted stoage log veso) taos/docs/ HDS-logpdf, Jauay 0 [5] D S Hochbaum ad J Geoge Shathkuma Covex sepaable optmzato s ot much hade tha lea optmzato J ACM, 37:843 86, Octobe 990 [6] M Gat ad S Boyd CVX: Matlab softwae fo dscpled covex pogammg, veso Apl 0 [7] M Gat ad S Boyd Gaph mplemetatos fo osmooth covex pogams I V Blodel, S Boyd, ad H Kmua, edtos, Recet Advaces Leag ad Cotol, Lectue Notes Cotol ad Ifomato Sceces, pages 95 0 Spge-Velag Lmted, 008 [8] S M Stefaov Covex sepaable mmzato subject to bouded vaables Comp Optmzato ad Applcatos, 8, 00