Nokia Research Center

Transcription

1 Noka Researh Ceter Sprg 24! % & ) )*+, -.% "#$ "#'( "#$ "

2 Itroduto 1 Itroduto Ths paper dsusses the state of the projet for ourse Mat Projet Semar Operatos Researh, ordered ad supported by Noka Researh Ceter. The teto s to gve a realst vew of the work doe so far, shfts projet fous, a bref rollout pla ad updated revew o projet s rsks. 2 Preset state of the projet Startg pot for the projet was more hallegg tha the team frst ojetured. Projet team dd ot reeve ay orete ad strtly delmted researh problem from ts let NRC. Istead, NRC provded us a more geeral top of terest. The pratal approah of how to go about the top ad to formulate a projet problem out of t was left to the projet group. Possble solutos to the problem ad relevat questos shaped durg the meetgs wth Noka s otat perso ad professor Salo as well as durg lterature study. Ths proess took more tme tha the projet team tally assessed ad also aused some hages to the projet pla. Itally t was teded to dede upo approprate ases, from whh a mathematal model ould be derved to fd out f t would be luratve to deompose the seleted omputatoal problems. Restruturg the problem led to the aalyss of defg the lmts to spefy whe t would be reasoable to osder adaptg grd omputato. Approprate ase problems wll be seleted after ths task s ompleted. Thus, projet s fous has shfted form the deep aalyses of partular ases to the formulato of parametrzed mathematal model. Aother hage, although a mor oe, was projet team s deso to alloate more resoures to wrtg the fal report. The projet group a, after defg the atual projet fous ad sope, state that the projet s o shedule. Lterature study over the subjet matter s ompleted ad the relevat ssues o the Noka s top of terest have bee uderstood. Data oerg requred model parameters s gathered. Ths ludes the tehal formato o the dfferet omputg deves ad possble ommuato lks betwee these deves. Noka Researh Ceter 2 Daelse, Hytöe, Kuusela & Ylä-Attla

3 How to otue Projet team has resheduled the ase seleto task to start at week 13 whe the mathematal model should be ready. Correspodgly, the struturg of mathematal model has bee partly ompleted. Results related to the model are desrbed more detal appedx A. Projet group wshes that our oppoets ould have a ostrutve ad rtal look over our model. 3 How to otue The projet group must thoroughly exame f the formulated mathematal model has a aalytal soluto. If the aalytal soluto exsts the oeto arhteture of the omputg deves a be expaded to lude star topology wth brahg odes.e. a tree topology. If the aalytal soluto does ot exst, we have to abde by the urret verso of the model ad settle wth the smple star topology. It s possble to evaluate the model wth approprate ase problems ether aalytally or umerally ad ths s doe to the extet of a reasoable workload. Oe ase that has already partly bee tested s the overso of vdeo sgal. Wrtg the fal report s also to be started soo. Noka Researh Ceter 3 Daelse, Hytöe, Kuusela & Ylä-Attla

4 Revew o the projet s rsks 4 Revew o the projet s rsks The major problem has bee fdg the balae betwee the futoalty ad outrght usefuless of the mathematal model ostruted. The model should be adequate desrpto of the doma of grd omputg as well as a hady tool for aalyzg deompostos of seleted grd omputg problems wth the dfferet arrays of omputg uts ad arbtrary star arhtetures. Noka Researh Ceter 4 Daelse, Hytöe, Kuusela & Ylä-Attla

5 Ths seto deals wth the questo what s a optmal soluto strategy a grd etwork for a perfetly deomposable problem. More formally, soluto strategy meas fdg a alloato of a problem amog deves that mmzes the tme T whh a problem P a be solved a grd etwork. The topology assumed s a star topology (Fgure 1). Later o we hope to expad ths to a tree topology. Fgure 1. Deves foud homes oeted a star topology. The Grd Evromet I ths part we assume oly a star topology for the grd etwork, lke the oe see Fgure 1. The etwork ossts of +1 omputg uts, where the etral deve orrespods to dex zero ad the surroudg deves orrespod to des 1,..,. Eah ut the grd has a omputg power. Betwee the etral deve ad eah surroudg deve there exsts a symmetr ommuato lk wth badwdth s ad latey l. Noka Researh Ceter 5 Daelse, Hytöe, Kuusela & Ylä-Attla

6 About the Problem I ths evromet we have a problem P, whh we a solve o the host omputer of the problem (assumed to be the etral ut) or a dstrbuted fasho the grd etwork of deves. The problem has a omplexty Y, a desrpto, whh has a sze X (later refered to as sze or desrpto merely), ad a aswer, whh has a sze R (later refered to as aswer merely). We assume that the omplexty a be dvded to wato depedet parts,.e. Y = y (1) ad we all ths kd of problem perfetly deomposable. Eah sub problem p wth the omplexty y has a desrpto x ad a aswer r. Objetve ad Costs The objetve s to fd the mmum tme take to solve the desrbed problem by dstrbutg the problem to dfferet deves. The osts urred whe solvg a problem a dstrbuted fasho ossts of deomposto osts, solvg osts ad omposto osts. The followg wll dsuss the osts urrg whe dog ths. Deomposto ad Composto Costs Before a problem a be solved a dstrbuted fasho the problem must be deomposed to depedet solvable parts of postve szes (x,,x ). T Deompose s the ost urred or tme take to perform the deomposto. After deomposg the problem, t a be solved the grd, whh takes a tme T Solve. Ths ost wll be dsussed more detal the ext hapter. As all depedet parts of the problem have bee solved, all depedet aswers must be omposed to the omplete aswer, whh takes tme T Compose. Thus, the tme take to solve a problem s take to be the sum of the deomposto ost, the solvg ost ad the omposto ost. Solvg Costs The tme take to omplete the th sub problem takes tme t. The tme t has a dfferet omposto or struture for the host deve ad for the surroudg deves. Before lookg at the dvdual duratos of ompletg dvdual sub problems we defe Noka Researh Ceter 6 Daelse, Hytöe, Kuusela & Ylä-Attla

7 the tme take to sed a desrpto over a ommuato lk, the tme take to ompute a dvdual sub problem, ad the tme take to sed a dvdual aswer problem over a ommuato lk. The tme take to sed a dvdual desrpto over a ommuato lk s defed as x t = + l sed, (2) s ad smlarly the take to sed a dvdual aswer over a ommuato lk s defed as r t = + l reeve, (3) s As the latees are eglgble omparso to the trasmsso tmes, the problems we wsh to solve, we wll drop latees l from the aalyss from here forth. Fally, the tme take to ompute a dvdual sub problem s defed as t = y ompute, (4) Tme take to solve sub problem p o the host omputer The host deve may start to solve the sub problem p mmedately after the deomposto of the problem s doe. The tme t ossts of the omputg tme o the host deve. Stll, the sedg ad reevg of data by the host omputer results the CPU tme beg dvded betwee omputg the sub problem p ad sedg ad reevg other sub problem desrptos respetvely aswers. We assume eah ommuato lk wll requre a peretage of the total omputg power deoted by. Thus, the tme take to solve p o the host s t = y ( x + r ) s (5) Tme take to solve sub problems p Noka Researh Ceter 7 Daelse, Hytöe, Kuusela & Ylä-Attla

8 We assume the ompleto of sub problem p a be dvded to three dsrete phases, amely, sedg of the sub problem over the ommuato lk, omputg of the problem p, ad sedg of the orrespodg aswer over the ommuato lk. Thus, the tme take to solve the sub problem p s t y = + = 1,, (6) x + r s Total solvg ost As we assume the omposto of the fal aswer a ot beg before all aswers have bee reeved by the host, the total solvg ost of the problem s {,..., } ( ) T Solve = max t (7) Smplfatos I order to derve a aalytal soluto we eed to make a few smplfatos. Frst we assume eah desrpto to be suh that X = x (8) Addtoally we assume there exsts a bjeto f that maps eah x oto y. Based o the above the futo f must be lear. More spefally we assume the depedey to be y = kx (9) where k s the omplexty oeffet or fator, whh s problem or algorthm depedet. Eah sub problem has a aswer, whh has a sze r, suh that R = r (1) ad r = x (11) Substtutg (9) ad (11) to (5) ad (6) yelds Noka Researh Ceter 8 Daelse, Hytöe, Kuusela & Ylä-Attla

9 t = kx ( 1 + ) s x (12) t = kx + ( 1+ β ) s x = 1,, (13) Furthermore we assume the deomposto ad omposto osts to be depedet oly o the total sze X of the problem. Hee, f we hoose to solve the problem a dstrbuted fasho the deomposto ad omposto osts may be omtted whe dervg the optmal alloato, as they do ot mpat t. O the other had the soluto derved ths maer must be ompared to the tme take to solve the problem oly o the host deve, before optmalty a be justfed. The optmzato As the total tme omputg ad data trasmt tme for deve s t, the total amout of tme eeded to ompute the whole problem s the maxmum of the dvdual deve tmes, as metoed. The task s to mmze the total tme eeded for the problem. Thus, the problem a be wrtte the form { t, t,..., t} m max 1 (14) If t s of the form 1 t ( b (15) x, x1, x2,, x ) = a x + a1x1 + + a x + ad the omputg tmes of other deves t,,t 2 1 t ( x ) = a x + b (16) The problem s therefore { t ( x, x, x,, x ), t ( x ), t ( x ),, t ( x )} m max (17) s.t. x = X 1 Equato (12) s of ths form whe b = 2 Equato (12) s of ths form whe b = Noka Researh Ceter 9 Daelse, Hytöe, Kuusela & Ylä-Attla

10 x =,, Wrtg { t } z = max (18) the problem a be formulated as m z (19) s.t. a x + a1x1 + + a x + b z a x + b z = 1,, x = X x =,, The above problem s to be solved wth the Karush-Kuh-Tuker optmalty odtos ad a aalytal soluto may be foud. Noka Researh Ceter 1 Daelse, Hytöe, Kuusela & Ylä-Attla