Distributed Load Balancing in a Multiple Server System by Shift-Invariant Protocol Sequences

Transcription

1 03 IEEE Wreess Communcaons and Neorkng Conference (WCNC): NETWORS Dsrbued Load Baancng n a Mupe Server Sysem by Shf-Invaran rooco Sequences Yupeng Zhang and Wng Shng Wong Deparmen of Informaon Engneerng The Chnese Unversy of Hong ong s076099@cuhkeduhk, song@ecuhkeduhk Absrac Ideay, many appcaon sysems for dsrbued users shoud be desgned hou requrng a cenrazed conroer, for exampe coud compung or reess sensor neorks A fundamena chaenge o deveopng dsrbued agorhms for hese sysems s oad baancng, hch s he focus of sudy n hs paper A common feaure of hese dsrbued agorhms s ha roung decsons shoud be dervabe hou requrng much nformaon from he sysem, probabsc roung s one exampe comng o mnd In hs paper, e propose a ne roung sraegy based on he dea of shf-nvaran prooco sequences We sudy hs oad baancng approach n he conex of a queung mode of mu-server sysem Our mode and sraegy can be apped o many pracca sysems, ncudng reess neorks Numerca sudes ere carred ou o compare our sraegy h oher roung sraeges such as probabsc roung and random sequences roung The resus sho ha he proposed agorhm has beer performance han hese sraeges eyords Wreess sensor neork; Mupe server sysem; Dsrbued oad baancng; Shf- Invaran rooco Sequences I INTRODUCTION Load baancng probem has been suded n many prevous orks Cho and oher [] presen a queung mode for a heerogeneous mupe server sysem In he mode, an arrvng job s roued by a job dspacher o one of parae servers Dfferen roung sraeges are aso suded n [], cassfed no o caegores: deermnsc and nondeermnsc In deermnsc sraeges, an ncomng job s sen o a parcuar server o mnmze or maxmze he expeced performance of a reaed creron funcon, such as mnmzng response me, mnmzng sysem me or maxmzng hroughpu In nondeermnsc sraeges, an arrvng job s sen o a server h ceran probaby, here he roung decson s usuay based on ndependen probaby dsncons erformance of hese sraeges are anayzed and compared n [] Cho and oher [] sho ha deermnsc sraeges have beer performance han nondeermnsc sraeges Hoever, n deermnsc sraeges, here mus be a cenra conroer o opmze her creron funcon and he compuaona compexy s usuay hgh, especay n sysems h arge number of servers On he conrary, n nondeermnsc sraeges, he packes are roued based on a probabsc roung marx, hch can be dsrbued o users ho are sendng jobs o he mupe server sysem and no cenra conroer s needed In hs paper, e are focus on he aer cass of probems, hch can be defned as dsrbued oad baancng N and Hang [4] propose a recursve probabsc roung agorhm I uses parameers of he sysem such as job ncomng rae and servce rae o adjus probabsc roung marx recursvey o fnay oban an opma marx Some oher prevous ork such as [6] and [7] are focused on ho o exchange nformaon among users and servers o dsrbue jobs eveny o a servers I can be veed as a rade-off beeen performance and nformaon In hs paper, a mode s presened o sudy he dsrbued oad baancng probem There are hree basc assumpons: () users ndependeny dsrbue her asks o servers accordng o pre-assgned bnary sequences, hch be expaned n foong secon; () no rea-me synchronzaon s requred; (3) here s no cenrazed conroer afer he na sequences assgnmen These assumpons usuay hod n reess neorks hou a cenrazed conroer or base saon and do no provde guaranee ha users are synchronzed Based on hese assumpons, a ne sraegy s proposed I can be shon ha he performance of he proposed sraegy s beer han probabsc roung The sraegy has many appcaons n pracca sysems For exampe, n some reess sensor neork, users ransfer daa hrough dfferen frequences As ong as severa users are ransmng on he same frequency, packe coson may occur and addona echnques such as soed aoha shoud be used o resove he conenon and he effecve ransmng me ncrease The dfferen frequences can be veed as dfferen servers and he exra ransmng me hen frequency coson occurs can be esmaed by queung mode Thus, he frequency aocaon n a reess sensor neork can be anayzed n he conex of oad dsrbuon n our mu-server mode, and our sraegy can hep dsrbue dfferen frequences o users eveny n order o reduce he ransmsson me Secon II presens he mode of dsrbued oad baancng probems and defnes he opmzaon funcon o make he The ork s suppored by a gran ened "Tme Crca Appcaons over a Shared Neork" of he Shun Hng Insue of Advanced Engneerng, The Chnese Unversy of Hong ong /3/$ IEEE 639

2 sysem performance robus Secon III nroduces he shf nvaran prooco sequences suded n [] and [3], and appes hem n he dsrbued oad baancng probems Secon IV shos he numerca resus and he comparson among dfferen sraeges Secon V concudes he paper II SYSTEM MODEL As shon n fgure, here are L servers n he mupe server sysem and users are sendng jobs o he sysem We assume ha each user sends he same number of jobs o servers each me so and e use N o denoe The paern for one user a one me so can be represened by a bnary sequence, here he posons of s denoe he servers seeced o send jobs o by hs user For exampe, f here are 6 servers, user sends jobs a me so o server and, e can use he sequence s (,, 0, 0, 0, 0) o represen he case Fg In order o dsrbue he jobs eveny, each user s requred o send jobs o dfferen servers n dfferen me sos, un a servers are used once Therefore, each user foo a seres of sequences o send jobs and he posons of s n hese sequences shoud no overap h each oher Afer usng a servers once, he user go back o foo he same paern of sequences and hus he seres s perodca We defne he perod as L / N and o smpfy he mode, e assume L s dvsbe by N We defne hs seres of sequences as S for user S { s, s,, s, s, s,, s,} () p Iusraon of mu-server sysem mode For exampe, suppose here are 6 servers and users and each user sends jobs every me so User seecs server and a he frs me so, server 3 and 4 a he second me so and server 5 and 6 a he hrd me so Ths scenaro can be represened as S {(,,0,0,0,0),(0,0,,,0,0),(0,0,0,0,,),} Smary, user may foo S {(,0,0,,0,0),(0,,0,0,,0),(0,0,,0,0,),} As here s no communcaon among dfferen users, jobs from dfferen users may arrve a he same server n one me so If more han one jobs are sen o he same server n a me so, e assume he queung order for hese jobs are deermned randomy In our mode, e ony consder he smpes case here one server serve exacy one job per me so Under hs assumpon, a saonary condon s ha N L and L / N Moreover, as he users are ndependen, he me o sar sendng jobs are no synchronzed and hus e need o nroduce a me dfference for each user hen he hoe sysem sars We defne he roaon RS of S by RS { s, s3,, s, s, s, s3,} () Then e nroduce he concep of a me dfference [0, ] for user so ha hen he sysem sars, user foos roaons of S R S { s, s,, s, s, s,} (3) In he prevous exampe, f 0 and 0, o users jus foo he sequences n S and S Hoever, f 0 and, he sequences for user become {(0,0,,0,0,),(,0,0,,0,0),(0,,0,0,,0),} because user aready goes o he hrd sequence n S hen he sysem sars Therefore, dfferen combnaons of (,,, ) resu n dfferen queung paerns for servers For a parcuar combnaon of, e can deermne he queung paern for a servers by he foong noaons We use s () o represen he vaue of poson n s I s jus he number of job user send o server a me (eher or 0) Noe ha s () s dependen of, and e use s () nsead of s (,,,, ) here for shor from; noaons of A() and Q () hch are defned aer shoud be nerpreed smary As one user ony sends one job o a parcuar server n one perod, e have s ( ) s ( ) s ( ) (4) One user sends N jobs each me so n oa o a servers n one perod, so s ( ) N (5) We furher use A () o denoe he number of jobs server receves from a users a me, hus A () can be compued drecy from s () A ( ) s ( ) (6) Smary, e have A ( ) A ( ) A ( ) ; (7) A ( ) N (8) As s () s perodca, A ( ) A ( ) (9) 640

3 Q () s defned as he number of jobs queung n server a me, can be expressed by he foong formua, provded ha e e ne jobs come and ge served mmedaey, hen compue Q () by Q ( ) ( Q ( )+ A ( ) ) (0) Q () s aso perodca excep for he frs me sos The proof of hs s gven n Appendx A Tha s, Q ( ) Q ( ) hen () Fnay, e can defne an average ang me for a random job by T Q (,,,, ) (,,, ) N () As Q () s compued under one parcuar combnaon of shfs as defned by, T(,,, ) can be represened as a funcon of We can use T(,,, ) o evauae he performance of he hoe sysem and a desrabe goa s o mnmze T(,,, ) Obvousy, T(,,, ) can assume dfferen vaues for dfferen combnaons of, for a fxed se of sequences aocaed o users Moreover, e canno predc he me dfferences for a users n pracce Therefore, e ry o fnd a se of sequences o mnmze he possbe maxmum vaue of T(,,, ) among a combnaons of me dfferences Tha s, our goa s o deermne he arg mn(max( T (,,, ))) (3) S, S,, S For exampe, suppose S {(,,0,0,0,0),(0,0,,,0,0),(0,0,0,0,,),} and S {(0,0,,,0,0),(0,0,0,0,,),(,,0,0,0,0),}, hen and 0, no jobs go o he same server a any 0 me so and hus he ang me T (,) 0 Hoever, hen and 4, he sequences ha o users foo are exacy he same and he ang me becomes much orse T (, 3) 05 To deermne he opma sraegy, e ony need o consder he ors case for each sequence se For exampe, n hs exampe, he ors case over a shfs s T (,3) 05 and hs se of S and S s no he opmum The proof s n Appendx B Based on hs heorem, as he sum of T(,,, ) s consan, a shf nvaran sequence se ha yeds he same vaue for a, f exss, s opma In he nex secon, e sho ha such souon exss for some sysem parameers III SHIFT INVARIANT ROTOCOL SEQUENCES rooco sequences are frs proposed n [] o be apped n coson channe hou feedback n reess communcaon Users are arranged o send packes based on he bnary sequences aocaed o hem If here are more han one packe sen o a recever smuaneousy, coson occurs and hese packes are dropped For users ha canno be synchronzed, he se of sequences shoud ensure ha no maer ho he sequences are shfed, he number of cosons, defned as cross correaon of he sequences, shoud be he same Ths knd of prooco sequence s caed shf nvaran prooco sequences (SIS) The condons of he coson channe s que smar o he dsrbued oad baancng mode, so e ry o appy he prooco sequences o sove he opmzaon probem n our mode As he cross correaon of shf nvaran prooco sequences are he same among dfferen shfs, and he response me s reaed o he cross correaon, e fnd ha shf nvaran prooco sequence s one of he sequence ses ha can make he response me equa among a shf paerns In oher ords, shf nvaran prooco sequence s one of he opma souons o (3) Fgure shos one se of shf nvaran prooco sequences and he resung roung paerns for dfferen users Tabe I shos he response me for a s, hch s consan We furher suded he case hen a users use he sequences for user 3 n prevous case Ahough he respond mes for some s are smaer han ha of shf nvaran sequences, he maxmum one s much arger Therefore, shf nvaran sequences se s one of he opma souons o (3) and hus has a robus performance To fnd he souon for he opmzaon probem (3), e rey on he foong heorem: Theorem: T (,,, ) C here C s a,,, consan for any sequences S, S,, S Fg exampe of shf nvaran prooco sequence 64

4 TABLE I AVERAGE RESOND TIME FOR DIFFERENT IV NUMERICAL RESULTS Hoever, he exsence of shf nvaran prooco sequences depends on he vaue of server number L and perod For users, he mnmum vaue of L s When he number of servers does no sasfy he requremens of shf nvaran sequences, e propose a smpe ay o generae a subopma sequence se Noe ha f here are o sequence ses S and S h same user number and perod, suppose her server numbers are L and L, e can append S a he end of S and e hem shf n her on range h perod In hs ay, e consruc a ne sequence se for L L servers h he same user number and perod Therefore, f here s a shf nvaran sequence se SL for server number L, e can generae opma sequence se for n L (here n s an neger) by addng n S L ogeher Based on, for any server number L, e can frsy fnd he shf nvaran sequence se S for servers, and hen defne a mnma shf nvaran sequence se for L by addng n L S ogeher here n s he maxma neger ha sasfes L nl L For he remanng par, e curreny append a sequence se h same sequences for a users Fgure 3 shos an exampe For L = 30, = 3 and = 3, e fnd he shf nvaran sequence se for L = 7 and append he same sequences for L = 3 a he end of sequences aocaed o each user The sengs of our smuaon are he same as menoned n secon II There are 7 servers and 3 users n he sysem and each user sends 9 jobs o dfferen servers every me so We furher nroduce a parameer Ts o denoe ho ong a snge job needs o be served I s assumed o be one me so n prevous par and can vary from o /3 n our smuaons (Jobs no cumuae f Ts s ess han /3, because even f a server receves 3 jobs n one me so, can serve hem a n ha me so) We compue average response me for one job defned by () Three roung sraeges are compared n our smuaon: probabsc roung, random sequence roung and shf nvaran sequence roung In probabsc roung, each user seecs 9 dfferen servers ou of 7 h equa probabes among a servers n each me so o send jobs Users behave ndependeny among dfferen me sos and her roung sraeges are no affeced by prevous seecons In random sequence roung, sequence ses defned by () are generaed randomy and aocaed o users a he begnnng Users send jobs foong he sequence ses n one perod = 3 me sos Then ne sequence ses be generaed randomy n nex perod In shf nvaran roung, e aocae he shf nvaran sequences for L = 7, = 3 and = 3, as shon n fgure, o users nsead of random sequences n prevous case o do roung Fgure 4 shos he comparson resu The range ne s probabsc roung and he performance s he ors The sysem s even no robus as T s goes o and he average response me ends o nfny Damond ne represens he performance of random sequence roung We can see s average response me s arger han ha of shf nvaran sequences roung, hch s denoed by sar ne L = 7, = 3, = 3 Fg3 sequences aocaed o users (L = 30, = 3, = 3) Fg4 comparson of average response me In pracce, users usuay dvde one pece of job no severa pars and dsrbue hem o servers and he oa ang me s deermned by he par ha reurns aes To smuae hs, n prevous sysem e furher compare he maxmum respond me for jobs sen n one me so from one 64

5 user Fgure 5 shos ha he shf nvaran sequences roung performs much beer han ohers n hs case Fg5 L = 7, = 3, = 3 comparson of maxma response me n one me so V CONCLUSIONS In hs paper, e presen a mode of dsrbued oad baancng probem, hch can be apped o many appcaons ncudng frequency channe aocaon n reess sensor neorks We defne an opmzaon probem n order o mnmze he maxma response me for a combnaons of me dfferences among he users Under suabe echnca condons, e derve opma souons o he probem, hch are based on shf nvaran prooco sequences Numerca resus sho ha our agorhm performs beer han oher sraeges such as probabsc roung and random sequences Hoever, here are some maons on he number of users Furher nvesgaons are requred o dea h cases havng arbrary number of users n he sysem REFERENCES [] Y C Cho, and W H oher, Modes for Dynamc Load Baancng n a Heerogeneous Mupe rocessor Sysem, Transacons on Compuers, 8(5), (May 979) [] W S Wong, Ne rooco Sequences for Random Access Channes hou Feedback, IEEE Transacons on Informaon Theory, 53(6), (June 007), pp [3] W Shum, C S Chen, C W Sung, and W S Wong, Shf-Invaran rooco Sequences for Coson Channes hou Feedback, IEEE Transacons on Informaon Theory, vo 55(7), pp 33-33, (Juy 009) [4] LM N and Hang, Adapve Load Baancng n a Mupe rocessor Sysem h Many Job Casses, IEEE Trans Sofare Eng, vo, no 5, pp , May 985 [5] Shubham Gupa, Transen Anayss of D()/M()/ Queung Sysem h Appcaons o Compung Arpor Deays, Maser Thess of Massachuses Insue of Technoogy, June 00 [6] A Barak and A Shoh, A Dsrbued Load-baancng ocy for a Mucompuer, Sofare-pracce and Experence, VOL 5(9), (Sepember 985) [7] rueger and R A Fnke, An adapve oad baancng agorhm for a mucompuer, Compuer Scence Deparmen, Unversy of Wsconsn, Madson, Wsconsn, 983 AENDIX A Consderng one parcuar server, I use Q () o represen he queue engh a me and A () o represen number of jobs arrve a me Defne D () as he number of jobs served a me (A me, ne jobs come frs, hen ge served mmedaey Q () s compued afer ha) Q( ) Q( ) A( ) D( ); (4) Q( ) A( ) 0 D ( ) 0 Q( ) A( ) 0 from (3), Q( ) A( ) D( ) Q(0) (5) A(), f Q(0) 0 and Q( ) b, hen D() b As D () s eher or 0, ( b) of D () are 0s and ohers are s D( ) D( ) D( ) 0 ), e can dvde he scenaro no severa Le ( b) ( ( b ) pars from (4), D( ) 0 Q( ) A( ) 0 and Q( ) A( ) D( ) Q(0) 0 Q( ) A( ) 0 D( ) because ony D( ) D( ) D( ( b) ) 0 smary, D( ) 0 Q( ) A( ) 0 and Q( ) A( ) D( ) Q( ) 0 Q( ) A( ) 0 D( ) because ony D( ) D( ) D( ( b) ) 0 No, I change he orgna condon Q(0) 0 o Q'(0) b and consder he same pons as before, Q'( ) A( ) D( ) Q'(0) b Q'( ) b D'( ) and (because A () does no change, D () canno be arger) smary, 643

6 , Q'( ) A( ) D( ) Q'( ) b Q'( ) b As, b ( b), D'( ) and b b b, D '( b) b b Q'( ) A( ) D( ) Q'( ) b Q'( ) 0 b Q'( ) Q'( ) Q'( ) 0 b b ( b) and exacy b me pons hen D ( ) 0 become D'( ) for ( b) and herefore, D'( ) Q'( ) Q( ) D( ) D'( ) Q'(0) Q(0) b b 0 Q'( ) Q( ) b AENDIX B We can compue he average ang me T(,,, ) from anoher drecon We defne as he me so a hch server receves a job from user n one perod, hch means s ( ),,,, The jobs sen o server n one perod can be represened by he sequence (,,, ) and A () can be compued by counng he number of s ha appear n (,,, ) Q () can furher be compued from A () by (8) Therefore, he oa ang me for jobs on server s a funcon of T (,,,, ) g(,,, ) (6) T (,,, ) can be furher compued by T (,,, ) (7) Therefore, T (,,,, ) L,,,,,,,,, T (,,, ) g(,,, ) T (,,,, ) L L For exampe, f S {(,,0,0,0,0),(0,0,,,0,0),(0,0,0,0,,),} and S {(0,,0,0,,0),(0,0,,0,0,),(,0,0,,0,0),}, hen 0 and 0, server receve one job from user a he frs me so and one job from user a he hrd me so, hus and 3 As user send exacy job o server n one perod, e can aays fnd a here user sends he job o server a he frs me so, hch means Then f, In hs ay, as he doman sze of and are boh, e can aays fnd exacy one correspond o one vaue of Therefore and are one o one mappng Thus, summng up from o s he same as summng up from o Meanhe, me dfferences for dfferen users are ndependen and s ony affeced by, e have g(,,, ),,, L g(,,, ) L g(,,, ) L Whch s he same no maer ha sequences are used For exampe, f S {(,,0,0,0,0),(0,0,,,0,0),(0,0,0,0,,),} and S {(0,,0,0,,0),(0,0,,0,0,),(,0,0,,0,0),}, hen e fx and ony consder server, e can fnd ha hen 0, hen and 3 hen Therefore, hen summng up a possbes for, s he same as summng up a possbe vaues for Smary, a vaues of are ncuded once hen summng up aerns on oher servers can aso be compued n hs ay and hus T T g, (, ) (, ) (, ) [ g(,) g(,) g(,3) g(,) g(,) (8) g(,3) g(3,) g(3,) g(3,3)] When e change sequences o S {(0,,0,0,0,0),(0,0,,0,0,),(,0,0,,0,0),} and S {(,0,,0,0,0),(0,,0,,0,0),(0,0,0,0,,),}, e can fnd ha he reaonshp changes o hen, hen and 3 hen 0 Hoever, hen e sum and up, e s ge he same expresson as (8) and he resu s rreevan o ha sequences used Therefore, T(,,, ) C here C s a consan for any,,, sequences S, S,, S 644