Randomized Load Balancing by Joining and Splitting Bins

Transcription

1 Radomzed Load Baacg by Jog ad Spttg Bs James Aspes Ytog Y 1 Itoducto Cosde the foowg oad baacg sceao: a ceta amout of wo oad s dstbuted amog a set of maches that may chage ove tme as maches o ad eave the system Upo a ava of a ew mache, oe of the exstg maches gves some of ts oad to the ew mache; ad upo a depatue of a mache, t gves a ts oad away to oe of the exstg maches the system Such oad baacg schemes ca be modeed as a smpe game of og ad spttg weghted bs Each b coespods to a mache the system, ad the weght of the b epesets the amout of oad assged to the mache The ava of a ew mache coespods to a spt of a b, ad the depatue of a exstg mache s epeseted by og two bs We cosde what happes whe the os ad spts ae adomzed Whe the bs ae spt wth pobabty popotoa to the weghts, t s ot had to see that ths gves the same behavo as ufomy cuttg a g as [3, whch yeds a Oog oad facto fo bs Whee t s feasbe to bas the adom choce wth the weghts, t s atua to mpemet ufom spts, whee the spt b s samped ufomy Ths s a atua choce, fo exampe, a pee-to-pee system whee a ufom sampg mechasm s avaabe eg, [1,4 Despte ts smpe defto, aayzg the pefomace of ths atua oad-dstbuto mechasm s a otva tas I ths pape, we appy a ove techque based o vecto oms to aayze the oad baacg pefomace of ufom adom os ad spts We show that f oy spts wth o os Depatmet of Compute Scece, Yae Uvesty Suppoted pat by NSF gats CNS ad CCF Ema: aspes@csyaeedu State Key Laboatoy fo Nove Softwae Techoogy, Nag Uvesty, Cha Suppoted by the Natoa Scece Foudato of Cha ude Gat No ad No Ema: yyt@uedu 1

2 ae apped, the expected oad facto, the ato betwee the maxmum weght ad the aveage weght of the bs, s betwee Ω 05 ad O 074 We the study the pefomace of mxed os ad spts, ad show that the expected oad facto s O 1/ 1 og afte ateatvey appyg suffcety may os ad spts to a abtay ta oad assgmet of bs These esuts demostate that the good oad facto obtaed by [3 depeds stogy o the abty to pefeetay spt heavy-oaded bs Load baace the spt-oy pocess I ths secto, we aayze the pefomace of a spt-oy pocess If we et N t deote the umbe of bs at tme t, the pocedue ca be descbed as foows Itay, N 0 1: thee s oy oe b, whose weght s 1, ad the absece of os we w aways have N t t + 1 Let X t deote the weght of b afte t spts, fo each [N t {0 N t 1} The weghts of the N t bs afte t spts ae ductvey defed as foows: Itay, t0 ad X Fo ate tmes t, ufomy choose a b fom [N t 1 Let X t X t 1 fo 0 < ; et X t 1 Xt 1 fo + 1; ad et X t X t 1 1 fo + 1 < N t 1 I othe wods, each spt chooses a b ufomy at adom ad spts t to two equay-weghted bs 1 Lowe boud Due to the defto of the spttg pocess, evey b the system has a weght of the fom 1/, whee s some tege Thee ae oe o moe heavest bs, whose weghts ae equa to the maxmum weght The maxmum oad deceases by haf oce the ast heavest b s spt, ad that spt decty ceates two ew heavest bs ad may decty ceate moe f the maxmum oad dops to the weght aeady peset some bs The maxmum oad w ot decease futhe ut both of these two bs have bee spt, ad t w ot decease aga ut the ast of the secod b s chde have bee spt, ad so o Itutvey, by cocetatg o the ghtmost path ths tee, we ca obta a owe boud o the sze of the actua agest b whe eepg tac of at most two bs at a tme Ths s fomazed the foowg theoem

3 Theoem 1 Let X 0, X 1,, X [ E max 0 X 1 be the weghts of the + 1 bs afte spts It hods that Poof: Cosde the foowg pocedue Itay, the oy b X 0 0 s maed as speca Whe a speca b s spt, f t s the oy speca b, the the two ew bs ceated by the spt ae speca, ad f othewse the two ew bs ae o-speca Afte ay umbe of spts, thee s ethe oe speca b, o two speca bs wth the same weght Let W be the weght of ay speca b afte spts Note that the speca b s ot ecessay the heavest b, but ts weght s cetay a owe boud of the maxmum oad, e max X a owe boud o E [ W W We ow poceed to obta Let w E [ W Fo b {1, }, et w b be the pat of w cotbuted by the cases that thee ae b speca bs Fomay, deotg by B the set of speca bs afte spts, t hods that B {1, } ad w b v v P[W v B b By tota pobabty, w w 1 + w Let deote the th spt b, whch s chose ufomy fom [ If thee s oy oe speca b, the ethe thee was pevousy oy oe speca b ad t was ot spt, o thee wee pevousy two speca bs ad oe of them was spt, theefoe w 1 W 1 v B 1 1 B 1 v P v W 1 v B 1 B 1 v 1 1 [ P W 1 v B v [ P W 1 v B 1 v v 1 1 w w 1 Smay, fo the case that thee ae two speca bs, w W 1 v B 1 1 B 1 v P v W 1 v B 1 B 1 v 1 P[W 1 v B v 1 v v 1 w w 1 3 P[W 1 v B 1

4 Reca that w w 1 + w, thus we have the foowg ecuso: w w 1 + w 1 1 w w w 1 Sovg ths ecusve equaty wth the obvous ta codto that w0 1, we have that w 1 [ Theefoe E max X w 1 Uppe boud Let X be the vecto deotg the weghts of the bs afte some umbe of spts Let X p be the p -om of X The maxmum weght ca be theefoe epeseted as X Note that X X p fo ay p 1 Ths meas that we ca get a uppe boud o the maxmum weght by boudg the p om fo ay such p The foowg theoem s deveoped usg ths dea Theoem Let X 0, X 1,, X be the weghts of the + 1 bs afte spts Fo ay [ α 1, t hods that E max 0 X O 1 1 α /α I ode to boud the om of the oad vecto, we fst boud aothe quatty Let w [ E 0 X, α whee α 1 s a paamete The foowg emma gves a ecuso fo w Lemma 3 It hods that w α w 1 Poof: Suppose that s the b to spt, whch s ufomy samped fom [ Fom the ductve defto of X t fom X t 1, t hods that 0 X α By tota pobabty, w E 0 X 1 X α X 1 α + α E 0 X 1 X 1 α + 1 α 1 α + 1 α 1 X 1 X 1 α α, 4

5 whch by eaty of expectato, mpes that 1 w E 0 X 1 α α [ E X 1 α α w 1 We ow poceed to pove Theoem Poof of Theoem : t hods that w 1 1 [ 0 X Let w E, α whee α 1 Accodg to Lemma 3, 1 1 α w 1 Obvousy w0 1 Sovg the ecuso, w α 1 exp α 1 exp O1 1 1 α 1 1 exp O1 1 1 α O 1 1 α Fo α 1, the fucto fx x 1/α s cocave Accodg to Jese s equaty, [ E max 0 X E 0 [ E 0 X X α 1/α α 1/α O 1 1 α /α We mmze the above boud umecay usg stadad methods By settg α 4, the expected maxmum oad s O 058 ad the coespodg expected oad facto s O 074 5

6 3 Ateato of os ad spts It s a bt supsg to see that os ca actuay optmze the oad facto I ths secto we study a atua case mxed os ad spts: ateatey appyg o ad spt opeatos to ta popuato of 1 bs We w see that ths sequece yeds a o1 expected oad facto, whch s much bette tha the spt-oy case We assume that tay thee ae 1 bs {X 0 } [ 1 wth a abtay oad dstbuto At each tme t, whee the bs ae {X t } [ 1, a b chose ufomy at adom s spt to poduce weghts {Y t+1 } [ ; ad the a pa of bs chose ufomy ae oed to gve ew weghts {X t+1 } [ 1 The tastos ae fomay defed as foows Fom {X t } [ 1 to {Y t+1 } [ : Exacty the same as the spt pocedue defed Secto, except that hee we wte Y t+1 Fom {Y t+1 < s Let X t+1 X t+1 Y t+1 } [ to {X t+1 Y t+1 f othewse stead of the oga otato X t+1 } [ 1 : Ufomy choose two bs {, s} fom [, whee + Y t+1 s f ; et X t+1 Y t+1 +1 f s ; ad et We pove a uppe boud o the maxmum oad afte appyg suffcety may ateatg os ad spts The theoem s poved wth a om-based techque whch s sma to the oe used the poof of Theoem Theoem 4 The expected oad facto foowg a sequece of ateatg os ad spts s Oexp O 1/ 1 og the mt That s, fo suffcety age, wth ay choce of o-egatve X 0 wth X 0 1 1, t hods that [ m E max X t t O exp Poof: Fst, we eed the foowg techca emma Lemma 5 Let x [0, 1 be a vecto such that 1 x 1 Fo ay teges, such that 6

7 > > 0, t hods that 1 x 1 x 1 x 1 Poof: Fo ay x, x, ad ay teges, > 0, t hods that x theefoe x + x x x x 1 x + x x x + x x 1 x x x 1 x + x x 1 x 1 It hods that, x 1 x 1 x 1 0, 1 x x x + 1 x + x 1 x x x x 1 x + x x x + x x 1 We the poceed to pove the theoem We wte X X t s toduced Let > 1 be a tege Let w t espectvey Specfcay, ad u t ad Y Y t f o ambguty be the -om of X t ad Y t w t E [ X 0 ad u t E [ 1 0 Y Note that X t s fomed by og two ufomy adom bs Y t Codtog o that the two adom bs ae {, s} [, t hods that X 0 Y + Y s + {,s} 1 Y 1 Y Y Ys 1 7

8 By tota pobabty, w t [ {,s} [ 1 E 0 X due to Lemma 5 w t E {,s} [ [ 1 Y 0 u t [ 1 1 E 1 Y {, s} Due to 1, E 0 s 1 [ E Y Ys u t u t u t 0 s E E Y Ys Y [ 1 Ys E 0 [ 1 0 [ 1 0 Y 1 Y Y 1 s0 Y s u t 1 Due to Lemma 3, t hods fo the spt opeato that u t w t 1 3 Combg ad 3, we obta the ecuso w t w t w t w t wt 1 1 Sce we see a uppe boud, we ca theefoe assume that w t 1 1 t 1 w wthout oss of geeaty We show by ducto that whe oog, w t + 1 wt / fo a suffcety age t It s tva to see ths s tue fo 1 as w t 1 1 fo ay t Suppose the hypothess s tue fo 1 We ca ewte the ecuso as w t 1 1 t 1 w / fo suffcety age t Let w be the fxed pot that w w / 1 1 +

9 + 1 1 The wt w w t 1 w a suffcety age t, w t w t, thus w w + fte steps of t Theefoe fo +3 4/ 1 1 Accodg to Jese s equaty [, fo oog, [ m E max X t t m t E [ / X w 1/ O 1 1 By settg og, whch s deed wth oog, we have that [ m E max X t t O exp 1 O 1+1/ 1 og 4 Cocuso We aayze the pefomace of a vey atua adomzed oad baacg scheme: ufomy og ad spttg weghted bs We deveop a om-based techque fo aayzg ths smpe pocedue By appyg the techque, we pove sevea bouds fo the expected oad facto Specfcay, f we eep ufomy spttg the bs wthout og them, the expected oad facto s betwee Ω 05 ad O 074, howeve, f we ateatvey o ad spt bs, the expected oad facto coveges to O 1/ 1 og These bouds ustfy the tuto that the powe of beg adaptve to the cuet oads s esseta fo oad baacg tass, ad they aso show a somehow supsg pheomeo that os ca actuay hep oad baacg f such powe s ot avaabe Refeeces [1 S Datta ad H Kagupta Ufom data sampg fom a pee-to-pee etwo I 7th IEEE Iteatoa Cofeece o Dstbuted Computg Systems ICDCS 007, page 50, 007 [ G Hady, J Lttewood, ad G Póya Iequates The Uvesty pess, 1934 [3 D R Kage, E Lehma, F T Leghto, R Pagahy, M S Leve, ad D Lew Cosstet hashg ad adom tees I STOC, pages , 1997 [4 D Stutzbach, R Reae, N G Duffed, S Se, ad W Wge O ubased sampg fo ustuctued pee-to-pee etwos IEEE/ACM Tas Netw, 17: , 009 9