Load Balancing Problems for Multiclass Jobs in Distributed/Parallel Computer Systems

Transcription

1 322 IEEE TRANSACTIONS ON COMPUTERS VOL. 47 NO. 3 MARCH 998 Lod Bncng Proes or Mutcss Jos n Dstruted/Pre Coputer Systes Je L Meer IEEE nd Hso Ked Astrct Lod ncng proes or utcss os n dstruted/pre coputer systes wth gener networ congurtons re consdered. We construct gener ode o such dstruted/pre coputer syste. The syste conssts o heterogeneous host coputers/processors (nodes) whch re nterconnected y genery congured councton/nterconnecton networ wheren there re sever csses o os ech o whch hs ts dstnct dey uncton t ech host nd ech councton n. Ths ode s used to orute the utcss o od ncng proe s nonner optzton proe n whch the go s to nze the en response te o o. A nuer o spe nd ntutve theoretc resuts on the souton o the optzton proe re derved. On the ss o these resuts we propose n eectve od ncng gorth or ncng the od over n entre dstruted/pre syste. The proposed gorth hs two ttrctve etures. One s tht the gorth cn e peented n decentrzed shon. Another eture s spe nd strghtorwrd structure. Modes o nodes councton networs nd nuerc epe re ustrted. The proposed gorth s copred wth we-nown stndrd steepest-descent gorth the FD gorth. By usng nuerc eperents we show tht the proposed gorth hs uch ster convergence n ters o coputton te thn the FD gorth. Inde Ters Agorths councton networs dstruted/pre coputer systes nterconnecton networs od ncng ngeent utcss os nonner optzton perornce queung theory. INTRODUCTION B ALANCING the worod over dstruted/pre coputer syste s portnt to prove ts over perornce (e.g. en response te) [] [22] [23]. Here the od ncng s o-schedung pocy whch tes o s whoe nd ssgns t to snge host coputer/processor. Oten os n dstruted/pre coputer syste cn e dvded nto derent csses. For nstnce there y e ntur dstncton ong dt voce nd vdeo pcets. We y dstngush derent csses o os y ther dey unctons. In ths pper we de wth the od ncng proes or such utcss os n dstruted/pre coputer syste tht conssts o heterogeneous host coputers/processors (nodes) nterconnected y genery congured councton/nterconnecton networ s shown n Fg.. The od ncng s conducted y trnserrng soe os ro nodes tht re hevy oded to those tht re ghty oded or de or processng. For studyng the od ncng proe gener thetc ode s deveoped. In ths ode the nodes re nterconnected y genery congured councton/nterconnecton networ. The nterconnecton ong nodes consdered n ths pper s qute gener; t ncudes prty nd uy connected networs such s str tree esh networs nd hypercue-type nterconnecton [7] ut does not ncude us connecton whch s The uthors re wth the Insttute o Inorton Scences nd Eectroncs Unversty o Tsuu Tsuu Scence Cty Ir 305 Jpn. E-: e@s.tsuu.c.p. Mnuscrpt receved 2 June 996; revsed 28 Apr For norton on otnng reprnts o ths rtce pese send e- to: tc@coputer.org nd reerence IEEECS Log Nuer currenty the ony tton. There re sever csses o os ech o whch hs ts dstnct dey uncton n ech node nd n n the syste. Aso the sce o the nterconnecton networ y e qute rge (e.g. wth thousnds o nodes). Jos re ssued to rrve t ech node ccordng to n ergodc process (e.g. te-nvrnt Posson process). Ech node deternes whether os o css w e processed ocy or e trnserred to nother node or processng. In the tter cse there s n councton dey ncurred s resut o trnserrng the o nd sendng c ts response. Snce ech councton n s u-dupe chnne the n councton dey depends on the o ow rtes n two drectons o n. Aso or processng o n node there s node processng dey. Ths ode s used to orute the utcss o od ncng proe s nonner optzton proe n whch the go s to nze the en response te o o. Despte the copety o the proe n ths pper we derve nuer o spe nd ntutve theoretc resuts on the souton o the optzton proe. On the ss o these resuts we propose n eectve od ncng gorth or ncng the od over n entre dstruted/pre syste. The proposed gorth hs two ttrctve etures. One s tht the gorth cn e peented n decentrzed shon. In order to deterne the opt od ncng ech node successvey updtes ts od ncng pocy sed ony on the oc norton out tse the trc through ts dcent ns nd the od o ts neghorng nodes connected y ts dcent ns. Another eture s very spe nd strghtorwrd structure /98/$ IEEE

2 LI AND KAMEDA: LOAD BALANCING PROBLEMS FOR MULTICLASS JOBS IN DISTRIBUTED/PARALLEL COMPUTER SYSTEMS 323 e ore strghtorwrd nd eectve thn the Tntw nd Towsey gorths. Furtherore they studed the opt od ncng proe or utcss os n us congured dstruted coputer syste [4]. The snge css o od ncng proes n tree herrchc networ congurton nd n str nd tree networ congurtons wth two-wy trc were studed y L nd Ked [7] [8] [6]. Ths pper es n portnt generzton o these reted wors. The rest o ths pper s orgnzed s oows. In the net secton we orute the opt od ncng proes n utcss o dstruted/pre coputer syste. Necessry nd sucent condtons or opt od ncng re derved n Secton 3. Sectons 4 nd 5 present n eectve od ncng gorth nd the proo o convergence o the gorth. The nuerc eperents re gven Secton 6. Secton 7 concudes ths pper. Fg.. A dstruted/pre coputer syste. Notce tht the od ncng proe s oruted s nonner optzton proe. The we-nown FD (Fow Devton) gorth [8] cn so e pped to sove the proe. We copre the perornce o our proposed gorth nd the FD gorth. By usng nuerc eperents we show tht the proposed gorth hs uch ster convergence n ters o coputton te thn the FD gorth. In our ode we ssue tht node processng dey nd n councton dey re gven. We notce tht the node processng dey cn e deterned dptvey n schee sr to tht o Ross nd Yo [20] y consderng schedung n ech node. In ths cse our gorth cn so e peented dptvey when the rtes t whch os rrve eterny t nodes re sowy vryng. We pont out tht od ncng consdered n ths pper s sttc snce t does not depend on the current stte o the nodes. Dync od ncng poces [5] [6] [9] [9] [2] [25] [26] oer the possty o provng od dstruton t the epense o ddton councton nd processng overhed. The overhed o dync od ncng y e rge [25] [26] n prtcur or rge heterogeneous dstruted syste. Coprng the perornce provded y sttc od ncng poces nd dync od ncng poces Ked et. [] nd Zhng et. [26] hve shown tht the sttc od ncng poces re preere when the syste ods re ght nd oderte or when the overhed s nonneggy hgh. The resuts o opt sttc od ncng y so hep us desgn dstruted/pre systes nd e pretrc dustent to prove the syste perornce. Soe reted wors hve een conducted or sttc od ncng proes or snge css os n dstruted/pre systes wth specc networ congurtons. Tntw nd Towsey [22] [23] studed opt sttc od ncng or snge o css n str nd us networ congurtons. On the ss o the Tntw nd Towsey wor K nd Ked [3] provded two proved gorths (K&K gorths) or sttc od ncng n str nd us networ congurtons respectvey whch re shown to 2 MODEL DESCRIPTION AND PROBLEM FORMULATION Consder dstruted/pre coputer syste s shown n Fg.. In the syste there re n nodes nterconnected y genery congured councton/nterconnecton networ. Ech node y contn one or ore resources such s CPU nd I/O devces contended or y the os processed t the node. The nodes n dstruted/pre syste re heterogeneous host coputers/processors; tht s they hve derent congurtons nuer o resources nd speed chrcterstcs. Let N e the set o nodes. E s set whose eeents re unordered prs o dstnct eeents o N. Ech unordered pr e < > n E s ced edge. For ech edge < > we dene two ordered prs ( ) nd ( ) whch re ced ns. Denote the set o ns y L. A neghorng node o the node s node whch s connected wth node y n edge. For node the set o neghorng nodes o node s denoted y V.e. V { ( ) ΠE}. There re csses o os n the syste. Denote C s the set o o csses. Css- os ( ΠC) rrve t node ( ΠN) ccordng to n ergodc process such s te-nvrnt Posson process wth the verge etern o rrv rte. A css- o rrvng t node y ether e processed ocy or trnserred through the networ to nother node or reote processng. In the tter cse there s dey ncurred s resut o trnserrng the o nd sendng c ts response. Denote s the rte t whch css- os re processed t node whch s so reerred to s css- od t node. Let e the css- o ow rte ro node to node so ced the css- trc on n ( ). The o ow n node s shown n Fg. 2. Notce n prtcur tht n the Tntw nd Towsey [23] ode o snge-css o dstruted syste they ssue tht o rrvng t node s cn ony e processed t node s ocy or e trnserred to neghorng node o node s or reote processng. Our ode s uch roder thn tht o Tntw nd Towsey s n the sense tht css- o rrvng t node s (s ΠN) y e processed t node s ocy or e trnserred through sever ns to node d (d ΠN) or reote processng.

3 324 IEEE TRANSACTIONS ON COMPUTERS VOL. 47 NO. 3 MARCH 998 the en councton dey G ( ) o n ( ) depends Fg. 2. Jo ow n node. Let y sd e the rte o css- os tht rrve t node s (s Œ N) (ncudng the css- os rrvng t node s eterny nd the css- os rrvng ro neghorng nodes o node s) nd re processed t node d (d Œ N). The od ncng pocy shoud ny deterne the o ow rte y sd. We notce tht once the css- od t ech node (" Œ N) nd the css- o ow rte on ech n ( ) ("( ) Œ L) re gven the css- o ow rte y sd s deterned under the condton tht the o ow nce constrnt s stsed (.e. ŒV V ). For epe the css- Œ o ow rte y sd s gven s oows. % s s d s s s K s 2 t- t s ysd & K ss s s s t- t K s > 0 > 0 t > 0 t > t 0 otherwse ' K where s s s the tot css- o rrv rte t node ŒV. Note tht n the cse s d y sd s n the epe. Hence we cn sy tht the od ncng pocy s deterned y the vues o nd where [ ] [ ] [ ] ( C ) [ ]. Let F ( ) e the en node dey (queung nd processng deys) o css- o processed t node where [ ] ( C ). Let G ( ) e the en councton dey o sendng css- o ro node to node nd sendng the response c ro node to node through n ( ). In prctce t s resone to ssue tht ony on the o ow rtes on n ( ) nd on n ( ) where [ ] nd [ ]. We use ( ) nsted o G ( ). Note tht the shpe o unctons F ( ) nd G ( ) y e copcted s shown n Secton 6. Assue tht the node dey uncton F ( ) ( Œ N) s derente ncresng nd conve uncton nd tht the councton dey uncton G ( ) (( ) Œ L) s postve derente nondecresng nd conve uncton. Note tht the css o dey unctons stsyng the ove ssuptons s rge nd t contns those o the M/G/ ode nd the centr server ode [] [5]. In the cse o the Ross nd Yo pper [20] the dey uncton s so proven to e ncresng nd conve. Let D( ) e the en response te o os verged over csses.e. the ente o spends n dstruted/pre syste ro the te o ts rrv unt the te o ts deprture. By usng Ltte s resut [5] D( ) s epressed s the su o the en node dey t nodes nd the en councton dey on ns s oows: G % ( 6 &K )K ŒC'K ŒN 6ŒL *K D F G F () where F Œ. N Œ C Our go s to nce the od whch s represented y the o-processng rtes nd the trnser rtes to nze the en response te. Other perornce esures such s weghted su o en response tes retve to os enterng t derent nodes y e consdered n sr shon. The proe o nzng the en response te o o s wrtten y consderng the structure o dstruted/pre coputer syste s oows: % &K 'K ŒC ŒN ŒL Mnze D F G F wth respect to the vres Œ C Œ N; Œ C ( ) Œ L suect to 6 ŒC ŒN ( )K *K (2) (3) ŒV ŒV 0 ŒN ŒC (3) 0 ŒL ŒC 6. (3c) The constrnt (3) s the ow nce constrnts ech o whch equtes the css- o ow rtes n nd out o node. Snce F ( ) ( Œ N Œ C) s n ncresng nd conve uncton nd G ( ) (( ) Œ L Œ C) s nondecresng nd conve uncton D( ) so s conve uncton.

4 LI AND KAMEDA: LOAD BALANCING PROBLEMS FOR MULTICLASS JOBS IN DISTRIBUTED/PARALLEL COMPUTER SYSTEMS 325 Notce tht en response te uncton D( ) s strcty conve we hve the unque souton to proe (2) [4]. Snce we do not ssue tht D( ) s strcty conve the souton y not e unque [2]. However we cn otn n opt souton tht nzes the en response te D( ) s w e shown shorty n the net secton though t y not e unque souton. 3 OPTIMAL LOAD BALANCING The oowng s study o the necessry nd sucent condtons or opt od ncng: Frst the rgn node dey uncton ( ) t node nd the rgn councton dey uncton g ( ) on n ( ) re ntroduced s oows: g D F F (4) ŒC FD (4) ŒC G G 4 9 ŒC 4 9. (4c) THEOREM. The set o vues o nd s n opt souton to proe (2) nd ony the oowng set o retons hod. > 0 ŒN ŒC (5) ŒN ŒC (5) g > 0 ŒL ŒC (5c) - g 0 ŒL ŒC (5d) suect to (6) ŒV ŒV ŒN ŒC where ( Œ N Œ C) s the Lgrnge utper. PROOF. To otn the opt souton to proe (2) we or the Lgrngn uncton s oows 6 6 ŒC ŒN ŒV ŒV H F D - -. (7) Dene the set o ese soutons o the ove optzton proe FS s oows: FS {( ) ( ) stses the constrnts (3)}. Note tht the set FS s conve poyhedron. Aso note tht the set FS s cosed ccordng to the constrnts (3) whch the o ow pttern ( ) n the set FS ust stsy. Snce the oectve uncton s conve y ssupton nd the ese set FS s conve set set o vues o nd s n opt souton to proe (2) nd ony t stses the oowng Kuhn-Tucer condtons (e.g. [0]) H (8) H H H (8) - 0 (8c) Œ N Œ C (9) g (9) g - 0 (9c) ( ) Œ L Œ C. 0. (0) ŒV ŒV - - ŒN The ove set o condtons (8) nd (9) re dentc wth the oowng set o retons: 2 7 > () 0 () Œ N Œ C (2) (2) - g > - g ( ) Œ L Œ C. The ove set o retons re equvent to the set n Theore. Accordng to the ove theore we hve the oowng resuts showng the propertes o the opt souton to proe (2). COROLLARY 2. The opt souton to proe (2) s cyce ree.e. there ests no css- n trc such tht > 0 > 0 > > 0 where re ndces o dstnct nodes n the networ. PROOF. It s proven y contrdcton. Assue there ests css- n trc n the opt souton such s > 0 > 0 > 0 > 0 where re ndces o dstnct nodes. Accordng to Theore we hve g - g (3)

5 326 IEEE TRANSACTIONS ON COMPUTERS VOL. 47 NO. 3 MARCH 998 Then g4 g g (4) whch contrdcts the ove ssupton tht the councton dey uncton G ( ) (( ) Œ L) s postve derente nondecresng uncton whch pes tht g ( ) > 0 or > 0. COROLLARY 3. In the opt souton to proe (2) the oowng retons hod true 0 > 0 " Œ L " ŒL 6 6. (5) PROOF. It s drect resut ro Corory 2. Ths corory ens tht n the opt souton the css- o ow rte ro the node to node nd the css- o ow rte ro the node to node cnnot e postve oth t the se te. Now we ntroduce soe dentons whch w e used n the render o the pper. Dene - ŒN ŒC. Notce tht ŒV- ŒV- ;@ ;@ y e nonnegtve ( 0) or negtve ( < 0). In the cse where s nonnegtve s the upper t o the css- o ow rte wth whch the css- os cn e sent ro node to ts neghor node. In ths cse cn e ten s the css- etern o rrv rte t node ( Œ V ) or node snce 0. I s negtve t ens tht the css- o ow rte ro node to node shoud e rger thn or equ to snce nd 0 n the opt souton. Note tht - 0 nd 0. We dvde the nodes n V the set o neghorng nodes o node nto the oowng our sets or ech css- n the opt souton: ) Set o css- ctve source nodes (A ): Node n the set sends prt o css- rrvng os to node nd so processes the other prt o css- os ocy.e. > > 0 - > 0; 2) Set o css- de source nodes (Id ): Node n the set sends css- rrvng os to node.e. 0; 3) Set o css- neutr nodes (Nu ): Node n the set does not send (or receve) ny css- os to (or ro) node.e. 0 ; 4) Set o css- sn nodes (S ): Node n the set receves soe css- os ro node.e. > 0 0. By usng Corory 3 we hve the oowng: COROLLARY 4. In the souton to proe (2) the sets A Id Nu nd S o ech css re utuy dsont nd V A Id Nu S " ŒC. (6). Thereore we hve the oowng theore: THEOREM 5. The set o vues o nd s n opt souton to proe (2) nd ony the oowng set o retons hod g g g 0 < < ŒA (7) - g ŒId (7) g > 0 ŒS (7c) 0 4 ŒNu 9 (7d) > 0 ŒN (7e) 0 ŒN (7) suect to ŒN. (8). ŒA ŒId ŒS 4 THE PROPOSED ALGORITHM An opt od ncng gorth or utcss os n dstruted/pre coputer systes s derved y usng Theore 5 ro the prevous secton. I we the vues o eeents o vectors (" Œ N) nd ("( ) Œ L) ecept the eeents (" Œ N) nd ("( ) Œ L) the dey unctons s we s the rgn dey unctons o css depend on eeents nd ony. In such cse we use ( ) nd g ( ) nsted o ( ) nd g ( ) respectvey. We otn the oowng retons or opt od ncng or css- os sed on Theore g g g 0 < < ŒA (9) - g ŒId (9) g > 0 ŒS (9c) 0 4 ŒNu9 (9d) > 0 ŒN (9e) 0 ŒN (9)

6 LI AND KAMEDA: LOAD BALANCING PROBLEMS FOR MULTICLASS JOBS IN DISTRIBUTED/PARALLEL COMPUTER SYSTEMS 327 suect to ŒN. (20) ŒA ŒId ŒS Due to the ssupton on F ( ) we note tht ( ) nd so then ( ) re ncresng unctons. The nverse o the rgn dey ( ) s dened s oows % nd > 0 &K 0 nd 0. 'K 4 9 Aso note tht g( ) s ether g( 0 ) or g( 0 ) snce 0 n the opt souton. Hereter we denote g( 0 ) y g( ) (( ) Œ L) n the opt souton. The retons (9) through (9) re rewrtten s oows: g g g 0 < < ŒA (2) - g ŒId (2) g > 0 ŒS (2c) 0 4 ŒNu9 (2d) ŒN. (2e) The ove retons py the oowng: I the vue o Lgrnge utper s deterned the set eers o A Id Nu nd S nd then the vues o nd (" Œ V ) re deterned unquey y usng retons (2) through (2e). The vue o s evuted y usng constrnt (20) s shown eow. For the se o convenence the oowng notton s ntroduced p - - g ŒV ŒN; q g ŒV ŒN. Cery p ( ) s onotoncy decresng uncton wth respect to nd q ( ) s onotoncy ncresng uncton wth respect to. The two unctons p ( ) nd q ( ) hve ther nverse unctons ( p ) - nd ( q ) - respectvey whch re dened n nner sr to the nverse uncton o. The oowng reton s otned y notng the denton o p nd q : q > q 0 p 0 > p > 0 " ŒV ŒN. (22) Note tht - Œ A nd Œ S. Accordng to retons (2) n Theore p Œ A q ŒS. Then (20) cn e rewrtten s oows: ŒA ŒId ŒS p q. (23) The ove equton pes tht the vue o cn e deterned unquey y spy usng one-denson serch ethod. The set eers o A Id S nd Nu re deterned unquey wth gven vue o s oows: J 4 9 J 49 Œ L 4 9 J J A p 0 < < ŒV (24) Id p V (24) S q > 0 ŒV (24c) L L Nu q 0 p 0 ŒV. (24d) An opt od ncng gorth tht soves the optzton proe (2) s derved y usng the ove retons. Ths gorth otns souton to optzton proe (2) gven n rtrry set o preter vues. $ The Lod Bncng Agorth ) Intzton. R 0. (r : terton nuer) Fnd ( (0) (0) ) (Œ FS) s n nt ese souton to the proe (2). 2) Copute the new vues (r) nd (r). Increse the terton nuer r y. Let (r) (r-) (r) (r-). For to ( C ) do the oowng or css- os (n Loop). egn F the eeents o vectors (" Œ N) nd ("( ) Œ L) ecept or the eeents (" Œ N) nd ("( ) Œ L). For to n (n N ) do the oowng or node (suoop). egn r Let ( ) r r ( ) r ( ) ( ) r ( nd ) (" Œ V ) e nd (" Œ V ) s dened n the sugorth gven eow respectvey. Appy the oowng sugorth to otn the r ( ) r ( ) r ( ) r ( new vues nd ) (" Œ V ). end end 3) Stoppng rue ( r) ( r) ( r-) ( r-) ( r) ( r) I D( )- D( ) D( ) < L e then STOP where e s proper cceptnce toernce; otherwse go to Step 2. $ Sugorth tht coputes the vues o " Œ V nd.

7 328 IEEE TRANSACTIONS ON COMPUTERS VOL. 47 NO. 3 MARCH 998 ) Ccute the vue o (" Œ V ). O(v) (v V ) -. ŒV-;@ ŒV-;@ (Note tht - -.) 2) Copre. O(v) I (" Œ V ) q ( 0) ( ) p ( 0) ths ens tht reton (2d) hods. In ths cse node n V eong to the css- neutr node the sugorth stops nd we hve 0 " Œ V nd. Otherwse proceed to Step 3. 3) Deterne nd the node prtton. O(v) Appy the goden secton serch (e.g. [24]) to nd such tht stses ŒA ŒId ŒS p q where the sets o A Id Nu nd S re deterned or gven s oows:. 4 9 J J J A p 0 < < ŒV Id p ŒV S q > 0 ŒV J Nu q 0 p 0 ŒV. L L L Proceed to the net step. 4) Deterne the vues o Œ V nd. O(v) 0 0 " ŒId p 0 " ŒA 0 q " ŒS " ŒNu. L In the gorth we hve to nd the nverse unctons ( p ) - nd ( q ) - whch y not hve cosed-or epresson. Genery the nverse unctons cn e esy otned nuercy y sovng nonner equton o snge vre. The proposed gorth hs two portnt etures. One s tht the gorth cn e peented n decentrzed shon. Notce tht the sugorth cn e peented n ech node nd tht t depends ony on oc norton (.e. node od nd n trc) coected ro neghorng nodes ony. Another portnt eture s the spe nd strghtorwrd structure. In ths gorth the rst step nds n nt ese souton ( 0 0 ). The second step coputes the new vues o ( r r ) tertvey y usng the sugorth. The n procedure o the sugorth s the goden secton serch used to deterne the vue o the Lgrnge utper. The st step o the gorth deternes whether the gorth terntes. Note tht the copety o the sugorth s O(v) where v s the nuer o neghor nodes o node (v V ). 5 CONVERGENCE OF THE ALGORITHM In ths secton we prove the convergence o the proposed gorth. Notce tht the gorth eecutes the sugorth tertvey or css 2... nd node 2... n. Tht s the sugorth s udng oc o the proposed gorth. Let us regrd the sugorth or node o css s n opertor : FS Æ FS where FS s the set o ese soutons. Dene nother opertor s coposton o opertors 2... n.. n $ n- $ $ In turn denote the proposed gorth y n opertor whch s coposton o opertors $ - $... $. Frst we show property o the opertor. Note tht the vues o vres ( Œ V Œ C) nd cn e chnged whe eepng the vues o other vres n nd constnt wthout votng the constrnts (3). I we eep other vres constnt the nzton proe (2) reduces to the oowng suproe: Mnze F J F 4 9 F G G U (25) ŒV wth respect to the vres ( Œ V ) nd suect to (26) ŒV ŒV ( )K *K - ŒV (26) 0 (26c) ŒV. (26d) (U n (25) denotes vre tht s ndependent o Œ V nd.) LEMMA 6. The opertor genertes the opt souton to suproe (25). PROOF: It s cer snce the opertor genertes the set o vues o ( Œ V ) nd whch stses the retons (7) through (7) n Theore 5.

8 LI AND KAMEDA: LOAD BALANCING PROBLEMS FOR MULTICLASS JOBS IN DISTRIBUTED/PARALLEL COMPUTER SYSTEMS 329 Deros nd Sprrow [4] hve derved soe eegnt resuts showng soe gener condtons or the convergence or n gorth. Here we ppy ther resuts to prove the convergence o the proposed gorth. Note tht s deterned s gven. Here we use D() nsted o D( ) or the se o convenence. Fro Deros nd Sprrow (reer to Theore (2.) nd Theore (2.2) n [4]) we hve the oowng e nd theore. LEMMA 7. I the set o ese soutons to proe (2) FS s cosed nd conve nd the opertor hs the oowng propertes then the proposed gorth gves n opt souton to proe (2). ) or soe ΠFS pes tht stses the retons gven n Theore 5 or ΠN nd ΠC. 2) s contnuous ppng ro FS to FS. 3) D() D() or ΠFS. 4) D() D() or soe ΠFS pes tht. THEOREM 8. The proposed gorth genertes sequence { (r) } whch converges to the opt souton to the optzton proe (2). PROOF. I the proposed gorth stses the propertes requred n Le 7 the proo o the theore cn e copeted. Now we prove tht hs the our propertes gven n Le 7. ) pes or ΠN nd ΠC. Tht s Property () n Le 7 hods. 2) Note tht ( ) s contnuous nd ncresng p ( ) s contnuous nd decresng nd q ( ) s contnuous nd ncresng. It s esy to see tht ( ΠC ΠN) s contnuous ppng ro FS to FS. Then s so contnuous ppng ro FS to FS y the denton. 3) Accordng Le 6 we hve D( ) D4 9. Notce tht nd then re coposton o opertors. Then we hve D() D( ) nd D() D() Tht s Property (3) n Le 7 stses. 4) I D() D() stses the opt condtons (7) through (7) gven n Theore 5. Tht s s the opt souton to proe (2). Then we hve. 6 NUMERICAL EXPERIMENTS Ths secton egns y studyng n opt utcss o od ncng n dstruted coputer syste y usng the proposed gorth. Then copre the gorth perornce o the proposed wth we-nown FD (Fow Devton) gorth [8] whch cn e pped to sove the opt od ncng proe. 6. An Epe o Opt Lod Bncng Consder dstruted coputer syste tht conssts o three host coputers (nodes) s ustrted n Fg 3. Fg. 3. An epe o utcss o dstruted coputer syste. Fg. 4. The utcss centr server ode o node. There re two csses o os n the dstruted syste. The centr server queung ode [5] s the coon ode or host coputer. Here the centr server ode s used s the node ode s shown n Fg. 4. In ths ode eponent server 0 odes CPU tht processes os ccordng to the processor shrng dscpne; eponent servers 2... d ode I/O devces whch process os ccordng to FCFS. Let p 0 nd p 2... d e the protes tht ter deprtng ro the CPU css- o eves node or requests I/O servce t devce 2... d respectvey. The epected node dey o o n such node ode s gven s d q F q q 0 d (27) where q0 p0 nd q p p0 nd s the css- processng rte o server 2... d t node. Note tht we ssue tht the schedung dscpne o the I/O servers ( 2... d ) s FCFS. We ssue tht 2 d or node n the node set N nd css n the set C so tht the BCMP theore [2] cn e pped. Te gves the preter vues o the node odes we use n ths nuerc enton. In ths prtcur

9 330 IEEE TRANSACTIONS ON COMPUTERS VOL. 47 NO. 3 MARCH 998 TABLE PARAMETER VALUES OF NODE MODELS Fg. 5. Node od nd n trc n opt od ncng. preter settng we otn the oowng en node dey unctons: F F F F F F (28) For the n councton ode consder the oowng cse: Assue tht the councton te or sendng css- pcet ro node to node hs n eponent dstruton wth en c (n.). Assue tht on the verge pcets re requred to descre css- o nd tht on the verge pcets re requred n sendng c response or css- o. Thus on the verge pcets re sent per nute ro node to node nd pcets re sent per nute ro node to node. Assue tht the en councton dey G ( ) ro node to node on n ( ) (( ) ΠL) s gven s oows: c c G4 9 c - - c c > c c c c ΠL (29) The vue o c n the cse whch we ene s gven n 6. Fg. 3. The cse where nd 2 20 s ened. By usng the proposed gorth t s shown tht the Fg. 6. An epe o dstruted coputer syste. en response te n the opt od ncng s n. whch s uch shorter thn the en response te.66 n. n the cse o no od ncng where ech node processes o ts own oc stre o os. The vues o node od nd n trc n the opt od ncng re ustrted n Fg Coprson o Agorth Perornce Notce tht the od ncng proe s oruted s nonner optzton proe. The we-nown FD (Fow Devton) gorth [8] cn so e pped to sove the proe. The FD gorth s stndrd steepest-descent gorth whch evutes ro ese souton the steepest descent drecton nd the step sze tht nzes the en o ow te ong such drecton to or new ese souton n ech terton. We copre the proposed gorth wth the FD gorth n ter o coputton te. For the se o spcty wthout oss o generty we consder snge-css o od ncng proe n dstruted coputer syste tht conssts o nne host coputers (nodes) connected v ner-neghor esh [7] congured networ s ustrted n Fg 6. In ths epe we seect the preters o host coputers nd councton ns sed on vues ound n estng coputer networs.

10 LI AND KAMEDA: LOAD BALANCING PROBLEMS FOR MULTICLASS JOBS IN DISTRIBUTED/PARALLEL COMPUTER SYSTEMS 33 TABLE 2 PARAMETERS OF NODE MODELS Processng rtes o Protes o o Jo rrv servers (os/n) evng CPU rte Node p 0 p p 2 p 3 p 4 (os/n) Fg. 7. Coprng perornce o two gorths n coputton te. The centr server ode s so used or the node ode. Te 2 gves the preters o the node odes we use n ths nuerc enton. Note tht there s ony snge css os n the syste. Thus the odes hve the oowng en node dey unctons: F27 F27 F27 5 < 20 or 6 9; 20-5 < 00 or ; 00-3 < 00 or 4 5. (30) 00 - For the n councton ode consder the oowng cse. Let the en councton te or sendng pcet ro node to node e /c (n.) wth the eponent dstruton. Assue tht on the verge there re pcets or o descrpton nd tht there re pcets or the response to the o. Sry s the cse n the ove epe we hve the oowng en councton dey G ( ) ro node to node on n ( ) (( ) Œ L) s oows: 4 9 G (3) c - c - c > nd c > ŒL where c s the n cpcty n pcets/nute the vues o whch re shown Fg 6. The cse where 00 nd 20 s ened. The coprson o the coputton te requreents s conducted nuercy y progrng these two gorths n Fortrn nd runnng the on Sprc worstton. The nuerc resuts re shown n Fg. 7. It shows tht the proposed gorth hs uch ster convergence to the nu response te thn the FD gorth. The ove nuerc resuts y e epected snce the proposed gorth hs uch sper structure. We hve otned sr resuts under derent syste odes wth derent preter settngs. 7 CONCLUSION We hve studed the opt od ncng proe n utcss o dstruted/pre coputer syste wth gener networ congurtons. We hve deveoped gener thetc ode or ths proe nd oruted t s nonner optzton proe. We hve derved the

11 332 IEEE TRANSACTIONS ON COMPUTERS VOL. 47 NO. 3 MARCH 998 necessry nd sucent condtons or the opt souton. On the ss o the study spe gorth hs een proposed to sove the gener od ncng proe. The proposed gorth s copred wth the we-nown FD gorth. It s shown tht the proposed gorth hs uch ster convergence speed thn the FD gorth n ters o coputton te. Here we woud e to enton soe etensons to ths wor. Adptve nd dstruted od ncng n whch the decson n node to trnser o depends on the stte o ts neghorng nodes w e n nterestng proe. Another etenson w e the study o the eects o syste preters (e.g. n councton te node processng te) on the opt od ncng. ACKNOWLEDGMENTS The uthors woud e to thn the nonyous revewers or ther vue suggestons tht proved the pper. REFERENCES [] A.O. Aen Proty Sttstcs nd Queueng Theory wth Coputer Scence Appctons second ed. Boston: Acdec 990. [2] F. Bsett K.M. Chndy R.R. Muntz nd F.G. Pcos Open Cosed nd Med Networs o Queues wth Derent Csses o Custoers J. ACM vo. 22 no. 2 pp Apr [3] S.C. Deros The Trc Assgnent Proe or Mutcss-User Trnsportton Networs Trnsportton Scence vo. 6 pp [4] S.C. Deros nd F.T. Sprrow The Trc Assgnent Proe or Gener Networ J. Reserch o Nt Bureu o Stndrds-B vo. 73B no. 2 pp [5] D.L. Eger E.D. Lzows nd J. Zhorn Adptve Lod Shrng n Hoogeneous Dstruted Systes IEEE Trns. Sotwre Eng. vo. 2 no. 5 pp My 986. [6] D.L. Eger E.D. Lzows nd J. Zhorn A Coprson o Recever-Intted nd Sender-Intted Adptve Lod Bncng Perornce Evuton vo. 6 pp [7] T. Feng A Survey o Interconnecton Networs Coputer pp Dec. 98. [8] L. Frtt M. Ger nd L. Kenroc The Fow Devton Method: An Approch to Store-nd-Forwrd Councton Networ Desgn Networs vo. 3 pp [9] M. Hrcho-Bter nd A.B. Downey Epotng Process Lete Dstrutons or Dync Lod Bncng Proc. 996 ACM SIG- METRICS pp Phdeph My 996. [0] M.D. Intrgtor Mthetc Optzton nd Econoc Theory. Engewood Cs N.J.: Prentce H Inc. 97. [] H. Ked J. L C. K nd Y. Zhng Opt Lod Bncng n Dstruted Coputer Systes. London: Sprnger-Verg 996. [2] H. Ked nd Y. Zhng Unqueness o the Souton or Opt Sttc Routng n Open BCMP Queueng Networs Mthetc nd Coputer Modeng vo. 22 nos. 0-2 pp [3] C. K nd H. Ked An Agorth or Opt Sttc Lod Bncng n Dstruted Coputer Systes IEEE Trns. Coputers vo. 4 no. 3 pp Mr [4] C. K nd H. Ked Opt Sttc Lod Bncng o Mut- Css Jos n Dstruted Coputer Syste Trns. IEICE vo. 73 no. 7 pp Juy 990. Fu ne o ourn? [5] L. Kenroc Queueng Systes Vo. 2: Coputer Appctons. New Yor: John Wey & Sons Inc [6] J. Lnd H. Ked Opt Sttc Lod Bncng n Tree Networ Congurtons wth Two-Wy Trc Coputer Networs nd ISDN Systes vo. 25 no. 2 pp [7] J. L nd H. Ked A Decoposton Agorth or Opt Sttc Lod Bncng n Tree Herrchy Networ Congurtons IEEE Trns. Pre nd Dstruted Systes vo. 5 no. 5 pp My 994. [8] J. L nd H. Ked Opt Lod Bncng n Str Networ Congurtons wth Two-Wy Trc J. Pre nd Dstruted Coputng vo. 23 no. 3 pp Dec [9] F.C.H. Ln nd R.M. Keer The Grdent Mode Lod Bncng Method IEEE Trns. Sotwre Eng. vo. 3 no. pp Jn [20] K.W. Ross nd D.D. Yo Opt Lod Bncng nd Schedung n Dstruted Coputer Syste J. ACM vo. 38 no. 3 pp Juy 99. [2] K.G. Shn nd Y. Chng A Coordnted Locton Pocy or Lod Shrng n Hypercue-Connected Mutcoputers IEEE Trns. Coputers vo. 44 no. 5 pp My 995. [22] A.N. Tntw nd D. Towsey A Gener Mode or Opt Sttc Lod Bncng n Str Networ Congurtons Proc. PERFORMANCE 84 E. Geene ed. pp North-Hond: Esever Scence Pushers B.V [23] A.N. Tntw nd D. Towsey Opt Sttc Lod Bncng n Dstruted Coputer Systes J. ACM vo. 32 no. 2 pp Apr [24] D.J. Wde nd C.S. Beghter Foundtons o Optzton. Engewood Cs N.J.: Prentce H 967. [25] Y. Zhng H. Ked nd K. Shzu Adptve Bddng Lod Bncng Agorths n Heterogeneous Dstruted Systes Proc. IEEE Second Int Worshop Modeng Anyss nd Suton o Coputer nd Teeco. Systes pp Durh N.C. Jn [26] Y. Zhng K. Hoz H. Ked nd K. Shzu A Perornce Coprson o Adptve nd Sttc Lod Bncng n Heterogeneous Dstruted Systes Proc. IEEE 28th Ann. Suton Syp. pp Phoen Arz. Apr Je L receved the BE degree n coputer scence ro Zheng Unversty Hngzhou Chn n 982 nd the ME degree n eectronc engneerng nd councton systes ro the Chn Acdey o Posts nd Teecounctons Beng Chn n 985. He receved the DrEng degree ro the Unversty o Eectro- Counctons Toyo Jpn n 993. Fro 985 to 989 he ws reserch engneer t the Chn Acdey o Posts nd Teecounctons Beng. Snce Apr 993 he hs een wth the Insttute o Inorton Scences nd Eectroncs Unversty o Tsuu Jpn where he s currenty n ssocte proessor. Hs reserch nterests re n coputer networs dstruted/pre systes nd odeng nd perornce evuton. He s eer o the IEEE nd the ACM. He s servng s nger o the Study Group on Syste Evuton or the Inorton Processng Socety o Jpn (IPSJ). Hso Ked receved the BSc degree n physcs n 965 nd the MSc nd DSc degrees or hs wor n coputer scence n 967 nd 970 respectvey ro the Unversty o Toyo Toyo Jpn. Snce then he hs wored t the Unversty o Toyo IBM T.J. Wtson Reserch Center nd the Unversty o Toronto the Unversty o Eectro-Counctons etc. Presenty he s proessor t the Insttute o Inorton Scences nd Eectroncs Unversty o Tsuu Jpn. He hs een conductng reserch on opertng syste desgn prncpes opertng syste schedung perornce esureent nd queung nyss o coputer systes dstruted nd pre processng etc. He hs pushed ppers n the Journ o the ACM ACM Trnsctons on Coputer Systes IEEE Trnsctons on Sotwre Engneerng IEEE Trnsctons on Coputers IEEE Trnsctons on Pre nd Dstruted Systes Opertons Reserch Act Inortc etc. He hs so een contnuousy nterested n gener systes pctons o coputer systes. He hs served s n edtor or ourns pushed y the Inorton Processng Socety o Jpn. He so served s the chrn o the Study Group on Queung Systes or the Opertons Reserch Socety o Jpn nd s the chrn o SIGOS (Opertng Syste) o the Inorton Processng Socety o Jpn. He s now the chrn o Study Group on Syste Evuton or the Inorton Processng Socety o Jpn.