Load Balacg va Rado Local Search Closed ad Ope systes A. Gaesh Dept. of Matheatcs Uversty of Brstol, UK a.gaesh@brstol.ac.u A. Proutere Mcrosoft Research Cabrdge, UK aproute@crosoft.co S. Llethal Stats Lab Cabrdge Uversty, UK s.llethal@statslab.ca.ac.u F. Satos INRIA Rocquecourt, Frace flora.satos@4x.org D. Majuath Dept. of EE IIT Muba, Ida daju@ee.tb.ac. ABSTRACT I ths paper, we aalyze the perforace of rado load resaplg ad grato strateges parallel server systes. Clets tally attach to a arbtrary server, but ay swtch servers depedetly at rado stats of te a attept to prove ther servce rate. Ths approach to load balacg cotrasts wth tradtoal approaches where clets ae sart server selectos upo arrval (e.g., Jothe-Shortest-Queue polcy ad varats thereof). Load resaplg s partcularly relevat scearos where clets caot predct the load of a server before beg actually attached to t. A portat exaple s wreless spectru sharg where clets try to share a set of frequecy bads a dstrbuted aer. We frst aalyze the atural Rado Local Search (RLS) strategy. Uder ths strategy, after saplg a ew server radoly, clets oly swtch to t f ther servce rate s proved. I closed systes, where the clet populato s fxed, we derve tght estates of the te t taes uder RLS strategy to balace the load across servers. We the study ope systes where clets arrve accordg to a rado process ad leave the syste upo servce copleto. I ths scearo, we aalyze how clet gratos wth the syste teract wth the syste dyacs duced by clet arrvals ad departures. We copare the load-aware RLS strategy to a load-oblvous strategy whch clets just radoly swtch server wthout accoutg for the server loads. Surprsgly, we show that both load-oblvous ad load-aware strateges stablze the syste wheever ths s at all possble. We further deostrate, usg large-syste asyptotcs, that the average clet sojour te uder the load-oblvous strategy s ot cosderably reduced whe clets apply sarter load-aware strateges. Persso to ae dgtal or hard copes of all or part of ths wor for persoal or classroo use s grated wthout fee provded that copes are ot ade or dstrbuted for proft or coercal advatage ad that copes bear ths otce ad the full ctato o the frst page. To copy otherwse, to republsh, to post o servers or to redstrbute to lsts, requres pror specfc persso ad/or a fee. SIGMETRICS 0, Jue 4 8, 200, New Yor, New Yor, USA. Copyrght 200 ACM 978--4503-0038-4/0/06...$0.00. Categores ad Subject Descrptors G.3 [Matheatcs of Coputg]: Probablty ad Statstcs Geeral Ters Queueg Theory, Mea Feld Asyptotcs, Stablty Aalyss. INTRODUCTION Load balacg s a ey copoet of today s coucato etwors ad coputer systes whch resources are dstrbuted over a wde area or across a large uber of systes ad have to be shared by a large uber of users. Load balacg eables effcet resource utlzato ad thereby teds to prove the qualty of servce perceved by users. Tradtoally, load balacg has bee acheved by applyg sart routg polces: whe a ew dead arrves, t s routed towards a partcular resource depedg o the curret loads of the varous resources, see [4] ad refereces there. I cotrast, we are terested systes where a ew tas s tally assged to a resource chose at rado rrespectve of the curret resource loads, but where tass ca be re-assged,.e., grate fro oe resource to aother. Our prary otvato stes fro the creasg popularty of Dyac Spectru Access (DSA) techques [] as a potetal echas for broadbad access future wreless systes. A coo pleetato platfor for DSA s the use of reprograable Software-Defed-Rados (SDRs). These ew rados are frequecy-agle or flexble, ad have the ablty to rapdly jup fro oe frequecy bad to aother order to explore ad explot large parts of the spectru. A cetral questo DSA s how ultple users ay farly ad effcetly share spectru a dstrbuted aer. Typcally, the servce rate of a user o a gve frequecy bad s versely proportoal to the uber of users trasttg o ths bad,.e., to the load of the bad. As ew users eterg the syste have o way of deterg the load o each frequecy bad, they have to tally select a bad radoly. Should a user receve a qute poor qualty of servce o a gve bad, she ay resaple a ew bad at rado ad decde to swtch to t. The overall syste perforace the strogly depeds o the dstrbuted resaplg ad swtchg strategy pleeted by each user.
Though our prary otvato s DSA, our ethods ad results could provde sght to a uber of other applcatos. Oe such pertas to wrele etwors, where there has recetly bee terest ultpath routg [5]. Here, users ay use several path to dowload fles, ad have to select the approprate path or the set of paths. Aother applcato s trasport etwors, where oe ght wsh to uderstad how Wardrop equlbra, whch correspod to the equalzato of jourey tes across alteratve routes, are acheved or approxated by etwor users actg o lted forato. Our results could also shed sght o how qucly such equlbra ca be re-establshed followg ajor dsruptos or other chages to the etwor. Fally, ote that dstrbuted load resaplg ca also be thought of as a gae betwee selfsh users. I fact, t s a stace of a cogesto gae (see e.g. [8]), ad our results shed lght o the te to reach a Nash equlbru such a gae, but t also helps uderstadg the outcoe of the gae wth a dyac populato of players. We cosder a geerc syste cosstg of ultple servers ( DSA, frequecy bads) eployg the Processor Sharg (PS) servce dscple, shared by clets who have to tally pc a server at rado, ad ay later resaple servers ad grate durg ther servce. We restrct our atteto to two atural dstrbuted resaplg ad grato strateges, the Rado Local Search (RLS) ad Rado Load-Oblvous (RLO) strateges. Whe pleetg the RLS algorth, a user resaples a ew radoly chose server at the stats of a Posso process, ad grates to ths ew server f ts load s saller tha that of the tal server. I cotrast, uder the RLO algorth, a user hops betwee servers accordg to a rado Marova jup process rrespectve of the loads of the vsted servers. We vestgate both closed systes wth fxed populato of clets, ad ope systes wth a populato whose dyacs are govered by clet arrvals ad the copletos of ther servces. I closed systes, we are terested characterzg the te that t taes uder the RLS algorth to balace all server loads (ote that here the RLO algorth does ot balace loads except a average sese so we do ot study ths algorth closed systes). I ope systes, users arrve at the varous servers accordg to depedet stochastc processes of fxed testes, ad leave upo servce copleto. I ths scearo, clet gratos wth the syste teract a coplcated aer wth the syste dyacs duced by clet arrvals ad departures. We a at characterzg syste stablty uder the RLS ad RLO strateges, as well as at dervg estates of user sojour tes. Our cotrbutos are as follows: Closed systes. We show that, startg fro a arbtrary allocato of users to servers, the te τ t taes to acheve perfect balace of server loads scales at ost as log()` 2 + log(), where ad deotes the uber of servers ad users, respectvely. Ths cosderably proves over the exstg bouds that stated that τ scales at ost as 2 (see e.g. [2]). We also vestgate the te τ ǫ to reach a approxate ǫ-balace (a syste reaches a approxate ǫ-balace f there exsts p such that the uber of users assocated to ay server les betwee ( ǫ)p ad ( + ǫ)p). Achevg such balace s uch faster tha reachg a perfect balace, ad we show that τ ǫ scales at ost as log()/ǫ. Ope systes. We deostrate that both RLS ad RLO strateges acheve the largest stablty rego possble,.e., that the syste s stable uder these two algorths provded that P λ < P µ, where λ deotes the tal user arrval rate at server ad µ s the servce rate of ths server. The result s ot surprsg for RLS, but less tutve for RLO sce, uder ths algorth, users tae o accout of server loads whe gratg. For both RLS ad RLO strateges, we derve approxate estates of the average user sojour te usg large-syste asyptotcs. The estates are show to be exact whe the uber of servers grows large, but tur out to be qute accurate for systes of lted szes as well. Our frst uercal results suggest that aga, surprsgly, the average clet sojour te uder the load-oblvous RLO strategy s ot cosderably reduced whe clets apply sarter load-aware RLS strategy. To our owledge, ths paper s the frst to aalyze the perforace of RLS ad RLO algorths ope systes. The paper s orgazed as follows. I the ext secto, we descrbe our odel ad otato. Sectos 3 ad 4 are devoted to the aalyss of closed ad ope systes, respectvely. We gve the related wor Secto 5, ad provde cocludg rears Secto 6. 2. MODEL DESCRIPTION AND NOTATION We cosder a set of Processor Sharg servers of respectve capactes µ,..., µ. The syste s hoogeeous f µ = for all =,...,. The syste state at te t s represeted by the uber of clets assocated to each server, N(t) = (N (t),..., N (t)). The servce rate of a clet assocated to server at te t s the µ /N (t). Clets depedetly resaple ad swtch servers to selfshly prove ther servce rate. They have a yopc vew of the syste the sese that they are aware of ther curret servce rates, but do ot ow the servce rate they would acheve at other servers. Gve ths yopc vew, t s atural to cosder ad aalyze the two followg rado dstrbuted resaplg ad grato algorths: Rado Local Search (RLS) algorth. At the stats of a Posso process of testy β > 0, a clet pcs a ew server uforly at rado ad grates to t f ad oly f ths would crease her servce rate. I other words, f at te t, a clet assocated to server pcs server j, she grates to j f ad oly f µ j/(n j(t) + ) > µ /N (t). Rado Load-Oblvous (RLO) algorth. After arrvg the syste, each clet vsts successve servers accordg to a cotuous-te rado wal wth trasto atrx Q = {q j,, j =,..., }. The rado wals are depedet across clets, ad rreducble. We deote by π the statoary dstrbuto of ths rado wal. Note that as a cosequece of rreducblty, clets vst all servers evetually,.e., π > 0 for all =,...,. Note that uder the RLO algorth, clets do ot tae loads to accout whe swtchg servers. I partcular they ay ove to a server wth a hgher load. A exaple of such a resaplg strategy s as follows. Each clet has
a Posso cloc of rate β > 0 ad, whe her cloc tcs, she pcs a ew server uforly at rado ad oves there rrespectve of ts load. We aalyze the perforace of dstrbuted resaplg ad grato strateges closed ad ope systes. I closed systes, the total populato of clets s fxed, equal to. For such systes, we vestgate the te t taes uder the RLS algorth to balace clets across servers, startg fro ay arbtrary syste state. I ope systes, exogeous clets assocate to server accordg to a Posso process of testy λ (the arrval processes are depedet across servers). Clet servce requreets are..d. expoetally dstrbuted wth ut ea. Uder RLS ad RLO algorths, (N(t), t 0) s a Marov process. I ope systes, we are terested characterzg the stablty rego of RLS ad RLO strateges, defed as the set of arrval rates λ = (λ,..., λ ) such that the syste s stable,.e., such that (N(t), t 0) s postve recurret. We also a at estatg the average clet sojour te. 3. CLOSED SYSTEMS I ths secto, we aalyze the perforace of the RLS resaplg strategy a closed hoogeeous syste, ad obta tght bouds o the expected te to balace the server loads. Recall that there are clets dstrbuted aog servers. Let = q+r, 0 r. We ow defe the followg: The state N(t) = (N (t),...,n (t)) s balaced f N (t) N j(t) for < j. The te to balace, τ, s defed as τ := f{t > 0 : N(t) s balaced}. The state N(t) s ǫ-balaced f ( ǫ)p N (t) ( + ǫ)p for all =,...,, where p = /. The te, τ ǫ, to ǫ-balace s defed as τ ǫ := f{t > 0 : N(t) s ǫ-balaced}. Let f, g : N R +. We say f() = O(g()) f there exst 0 N ad c R + such that f() cg() for all 0. Slarly, for f, g : N 2 R +, we say f(, l) = O(g(,l)) f there exst 0, l 0 N ad c R + such that f(, l) cg(, l) for all 0 ad l l 0. 3. Te to reach balace We ow characterze the te requred by the RLS algorth to reach perfect balace ad ǫ-balace. Theore 3.. The expected te, E[τ], for radozed local search to acheve balace s O(log()( 2 + log())). Theore 3.2. The expected te, E[τ ǫ] for radozed local search to acheve ǫ-balace s O(log()/ǫ). Rears. It s easy to see, applyg Marov s equalty, that the sae upper bouds o the te to balace hold probablty as expectato. 2. We ow copare our bouds o τ wth that fro [2]. Fro Theore 2.7 of [2], the expected uber of attepted oves before reachg balace s O( 2 ). Sce ove attepts (resaplg) occur at rate, ths gves us a te coplexty of O( 2 ). Our boud Theore 3. s uch tghter. 3. Our boud s close to the best possble. To see ths, suppose dvdes exactly. At soe stage, the algorth wll reach a allocato whch oe server has / + clets, oe other server has / clets ad all others have exactly / clets. Each of the / + clets at the overloaded server attepts to ove at rate, ad each ove attept s successful wth probablty /. Hece, the ea te for just the fal ove s 2 /( + ) 2 /(2). Our boud s oly a log factor hgher tha the te for the last ove. Alteratvely, cosder the stuato whe 2 = o() ad all clets are tally at the sae server. The, at least / clets eed to ove out of ths server to reach balace. Whe there are clets at the server, the expected te to the ext ove s at least / (possbly ore, as the ove attept ay ot be successful). Hece, the expected te to reach balace s at least = / + Z dx = log. / x Aga our boud s oly a log factor hgher tha the above lower boud o the te to reach balace. 3.2 Proofs Wthout loss of geeralty, we tae the rate β of the depedet Posso clocs at each clet to be uty. A clet at server whose cloc has tced at te t attepts to ove by saplg a server uforly at rado fro all servers. It oves to the sapled server, say j, f ad oly f N (t) N j(t) >. Clearly, N(t) evolves as a cotuous te Marov cha. 3.2. Proof of Theore 3. Defe V (t) := ax j N j(t),.e., V (t) s the axu uber of clets assocated wth ay server at te t. Defe C v(t) to be the uber of servers wth exactly v clets, B v(t) to be the uber wth exactly v clets ad A v(t) to be the uber wth strctly less tha v clets, all at te t. The dea of the proof s as follows. The evoluto of N(t) towards balace s dvded to phases. If V (t) = v, the N(t) s sad to be phase v. Thus, C v(t) s the uber of axally loaded servers phase v. Sce a clet ever oves to a server that has ore clets tha ts curret server, V (t) s ootoe decreasg ad, each phase, C v(t) s also ootoe decreasg. Phase v eds whe C v(t) = 0. Let τ v deote the (rado) legth of phase v. Each phase ca be further dvded to sub-phases, say (v, c), whe C v(t) = c. Let τ v,c deote the rado legth of te that t taes for C v(t) to decrease fro c to c. Observe that τ v = P c τv,c ad τ = P v τv. Whe N(t) s balaced, V (t) =, C (t) = r f r > 0 ad C (t) = otherwse. Ths gves us the axu rage for v. The uber of sub-phases each phase s also slarly bouded. The theore s proved by boudg the expected tes of each of the sub-phases ad phases.
Proof. I phase v, observe that vc v(t) + (v )B v(t), B v(t) C v(t) = A v(t). Further, for / v /( ), /(v ) (, ], but f N(t) s ot balaced, the there has to be at least oe server wth v 2 or fewer clets (.e., A v(t) ). Hece A v(t) ax o v,. () Each of the vc v(t) clets at oe of the axally loaded servers saples oe of the servers at rado at ut rate. If the sapled server happes to be oe of the A v(t) servers wth v 2 or fewer clets, the the clet oves to the sapled server ad C v(t) decreases by. Ths evet has probablty A v(t)/. Hece, C v(t) decreases by at a rate o saller tha vc v(t)a v(t)/ ad fro (), we obta that τ v,c τ v,c Exp(λ v,c), where λ v,c := vc Av(t) v(t) = vc ax (v ), o. (2) Here we wrte Y to ea that s stochastcally doated by Y, (.e., for all t, P[ > t] P[Y > t]), Y to ea that they have the sae dstrbuto ad Exp(x) to deote a expoetally dstrbuted rado varable wth rate x. I partcular, E[τ v,c] E[ τ v,c] vc (v ) o [(v ) ],, (3) + where x + deotes ax{x,0}. At ay te t, C v(t) s bouded above by /v, sce there caot be ore tha ths ay servers wth v clets. Sce phase v eds whe C v = 0, we have E[τ v] P /v c= τ v,c, ad we obta E[τ v] (v ) [(v ) ] +, o v /v c= c. (4) Fally, τ, the te t taes to reach perfect balace, satsfes τ τ v + τ /,c. (5) v= / + Now, we have by (3) that, c=r+ E[τ /,c ] c=r+ / 2 + For all v, we also readly see that, /v c= c c=r+ Z c x dx = 2 ( + log ). (6) Z /v + x dx + log v + log. Hece, fro (4), (5) ad (6), we obta /( ) E[τ] + log 2 + v v= / + + (v ). (7) v= /( ) + The uber of ters the frst su above s at ost ax{, }. Each suad s o ore tha ( ) 2 /. Hece, the frst su s bouded above by ax{ 2,2}. The secod su s bouded above by + = Z + ( ) (x ) dx + log ( ). Substtutg these expressos (7) ad splfyg, we get E[τ] ( + log ) ax{ 2 2,2} + +log( 2 ) + 2 2 3( + log ) + log +. Ths copletes the proof. 3.2.2 Proof of Theore 3.2 We eed the followg deftos. Let p = /. Server s ǫ-balaced at te t f ( ǫ)p N (t) (+ǫ)p, uderloaded f N (t) < ( ǫ)p ad overloaded f N (t) > ( + ǫ)p. M C(t), M U(t) ad M O(t) deote the uber of ǫ-balaced, uderloaded ad overloaded servers, respectvely. The uderflow fro server s defed to be ( 0 f N (t) p u (t) = p N (t) otherwse. Also, let U(t) := P u(t). Slarly, defe the overflow fro server as ( 0 f N (t) p o (t) = N (t) p otherwse, ad O(t) := P o(t). Proof. Let N O(t) be the uber of overflowg clets defed as N O(t) := (N (t) ( + ǫ)p), M O (t) where M O(t) s the set of overloaded servers at te t. We ca wrte U(t) O(t) Sce O(t) = U(t), we obta (M U(t) p) + M C(t) (ǫp), N O(t) + ( M U(t) M c(t)) (ǫp). p M U(t) + (ǫp)m C(t) N O(t) + ( M u(t) M C(t))(ǫp),
whch yelds N O(t) p(m U(t) + M C(t))( + ǫ) ǫ) N O(t) M C(t) + M U(t) ( + ǫ)p + ǫ + ǫ j ff NO(t) ax ( + ǫ)p, ǫ + ǫ. Now cosder a clet that s atteptg to ove at te t. We say that ths attept results a good ove f the attept results a grato that reduces N O(t). Let G deote the evet correspodg to a good ove. Whe the state of the syste s (N O, M C, M U), the probablty of a good ove s P(G) NO + ( + ǫ)p M C + M U ad the uber of attepts betwee successve good oves s geoetrc wth ea at ost (N O +p)m U. Let K G deote the uber of attepts before a good ove occurs fro the state (N O, M C, M U). The expected uber of attepts before a good ove reduces N O satsfes E[K G] (N O + ( + ǫ)p)(m C + M. U) Let K ǫ deote the uber of attepts to acheve ǫ-balace. I the worst case, N O(t) starts at ( )p ad eds at. We ca the boud E[K ǫ] as follows. E[K ǫ] = ( )p ǫ (( + ǫ)p + )(ax{ ǫ (( + ǫ)p + ) +ǫ + = + ǫ ǫ ( )p =ǫ+ ǫ (( + ǫ)p + ) (( + ǫ)p + ), ǫ }) (+ǫ)p +ǫ (+ǫ)p ( )p + ( + ǫ)p + ) =ǫ+ + ǫ «( + ǫ)p + ǫ log ǫ ( + ǫ)p «( )p ( + ǫ)p + ǫ + log ǫ ( + ǫ)p + ( )p ǫ log( + ǫ) + + + ǫ ǫ ǫ «ǫ log(). Sce each clet s saplg at ut rate, the total saplg rate s ad the average te to reach ǫ-balace, E[τ ǫ] s E[K ǫ]/. Thus E[τ ǫ] = O((log )/ǫ). 4. OPEN SYSTEMS I ope systes, we are terested quatfyg classcal queueg perforace etrcs, such as the stablty rego ad the ea clet sojour te. We frst vestgate the stablty rego acheved uder RLO ad RLS algorths. Both algorths are show to stablze the syste wheever ths s at all possble, whch for load-oblvous RLO algorth ay be surprsg. The, we try to obta ore detaled estates of the syste perforace. As t turs out, the syste equlbru dstrbuto s dffcult, f ot possble, to derve, ad we rely o large-syste asyptotcs to provde sghts to the way the syste behaves. 4. Stablty I the followg, we deote λ = (λ,..., λ ) ad µ = (µ,..., µ ) the vectors represetg the arrval ad departure rates at the varous servers. deotes the L -or o R. We frst provde a upper boud o the axu stablty rego defed as the set of λ such that there ay exst a resaplg ad grato strategy stablzg the syste. Ths set s obtaed by assug that all servers resources are pooled. Proposto 4.. Assue that λ s such that P P λ > µ. The there s o resaplg ad grato strategy stablzg the syste. Proof. The proof s straghtforward. Rear that for ay resaplg ad grato strategy, the total servce rate s less tha P µ. The f P λ > P µ, the average uber of clets the syste grows at a rate greater tha P P λ µ > 0. The syste s the ustable. The two followg theores state that both RLO ad RLS strateges acheve axu stablty. Theore 4.. Assue that P λ < P µ. The the syste s stable uder RLO algorth. Theore 4.2. Assue that P λ < P µ. The the syste s stable uder RLS algorth. A result soehow slar to that of Theore 4. was frst stated [7] usg heurstc flud lts arguets. Flud lts are powerful techques to study ergodcty of Marov processes [8]. They coprse the study of the syste behavor the followg ltg rege: the tal codto s scaled up by a ultplcatve factor, te s accelerated by the sae factor, ad teds to. Ofte the syste becoes tractable ths rege ad eve deterstc. If the syste the flud rege reaches 0 a fte te, the the process s ergodc. I the flud rege, clets stay for very log perods of te our syste, ad sce, uder RLO algorth, the clet rado wals are ergodc, the probablty that a gve clet s assocated to server should be proportoal to π (the equlbru dstrbuto of the rado wal). I such case, whe the clet populato s large (as the flud rege), all servers should be occuped ad actve, esurg that the syste eptes fte te. Ths s the arguet used [7], but ot justfed. The proble arses because the clet grato process actually teracts wth arrvals ad departures. Hadlg ths teracto turs out to be extreely dffcult. Recetly however, [2], the authors were able to forally derve the syste flud lts, ad aalyze ts stablty uder very specfc assuptos o the clet rado wal (ts trasto atrx Q has to be dagoalzable). Ther proof s qute trcate. I the followg, we prove Theore 4. wthout the use of flud lts, ad for ay rado wal. Our proof s uch ore drect tha that [2], ad hece s aeable to deal
wth ore geeral cases ad possble extesos. For the proof of Theore 4.2, we use a rather classc ethod,.e., we exhbt a sple Lyapuov fucto. 4.. Proof of Theore 4. Recall that by defto, uder RLO strategy, the process (N(t), t 0) s the Marov process wth the followg ozero trasto rates for j : 8 < : Ω(, + e ) = λ, Ω(, e + e j) = q j, Ω(, e ) = µ ½{ >0}, where = (,..., ) N ad e s the -th desoal vector wth every coordate equal to 0, except for the th oe equal to. The atrx Q = (q j) descrbes the grato of clets, ad t s oly assued to possess a uque statoary dstrbuto π = (π ) such that π > 0 for each =,...,. The a of the aalyss s to use the followg result, ow as Foster s crtero [20]. (Foster s crtero) If there exst K ad t 0 such that (8) sup E ( N(t) ) < 0, (9) N : K where E ( ) = E( N(0) = ), the (N(t), t 0) s ergodc. Kologorov s equato s the frst step that leads to (9): for ay t 0, the drft E ( N(t) ) s gve by! E ( N(t) ) = λ t Z t E 0 µ ½{N (u)>0} du. Ths gves the followg equalty, whch s the bass of our drft aalyss: E ( N(t) ) λ t µ Z t 0 P (N(u) > 0) du, (0) where P [ ] = P[ N(0) = ], ad for x N, x > 0 s to be uderstood coordatewse,.e., x > 0 for each =,...,. The dea of the proof of (9) s that whe the syste starts wth ay clets, the the uber of arrvals ad departures s eglgble o the te terval [0, t] ad the syste behaves le the closed oe. For a closed syste, t s ot dffcult to show, usg the fact that Q has a varat easure, that P(N(u) > 0) for u > 0 s arbtrarly close to as the uber of clets the syste creases. I vew of (0) ths gves a egatve drft whe P λ < P µ. The followg couplg tally proposed ad forally justfed [2] s ey to relate the ope ad closed systes. For N ad l, ρ R +, deote by N l,ρ the process uder RLO strategy startg the tal state, wth arrval rate l at server wth capacty ρ. The (N l,ρ(t)) s the Marov process wth N l,ρ(0) =, ad wth o-zero trasto rates gve by (8) wth l stead of λ ad ρ stead of µ. The the processes N l,0 ad N 0 ρ,0 ca be coupled such a way that for soe process Z(t) 0, N l,ρ(t) = N l,0(t) N 0 ρ,0(t) + Z(t), t 0. Moreover, the processes N 0 ρ,0 ad N l,0 are depedet, ad N 0 ρ,0 s a Posso process wth paraeter ρ. Essetally, ths couplg realzes the process N l,ρ wth arrvals ad departures as the dfferece betwee two processes wthout departures. Ths couplg ca be costructed as follows: cosder a partcle syste wth three ds of partcles, colored blue, red ad gree. All the partcles the syste are perforg depedet cotuous-te rado wals, gog fro to j at rate q j, ad the syste starts wth oly blue partcles. Cosder two depedet Posso processes N l ad N ρ wth respectve paraeters l ad ρ : at tes of N l, add a ew blue partcle at server wth probablty l / l. At tes of N ρ, cosder server wth probablty ρ / ρ : f there s a blue partcle, choose oe at rado ad tur t to a red oe. If there s o blue partcle, add a gree partcle. If B (t),r (t) ad G (t) are respectvely the uber of blue, red ad gree partcles at server at te t, the t s easy to see that: B s dstrbuted le N l,ρ, B + R s dstrbuted le N l,0, R + G s dstrbuted le N 0 ρ,0 ad B + G = N ρ s depedet of B + R. Ths proves the couplg wth Z(t) = G(t). The process N l,0 ca be see as the superposto of the tal partcles wth the partcles arrvg at rate l, hece the addtoal couplg N l,0 = N 0,0 + N 0 l,0 holds, ad fally, N l,ρ ca be wrtte N l,ρ(t) = N 0,0(t) + N 0 l,0(t) N 0 ρ,0(t) + Z(t), t 0, wth N 0,0 ad N 0 ρ,0 depedet, ad Z(t) 0. Startg fro (0), we ow tur our atteto to provg the exstece of costats K ad t whch satsfy (9). We have, usg the couplg s otato, P (N(u) > 0) = P(N λ,µ(u) > 0) ad hece, for ay 0 u t ad N, P (N(u) > 0) = P(N 0,0(u) + N 0 λ,0(u) + Z(u) > N 0 µ,0(u)) P(N 0,0(u) > N 0 µ,0(u) ). Sce the process N 0,0 s depedet of the rado varable N 0 µ,0(t), we ca wor codtoally o the value of N µ,0(t) ad study the quatty P(N 0,0(u) > M). Thus we oly eed to cosder the closed process N 0,0 heceforth, ad so we splfy the otato ad ote N 0,0 = N. Marov s equalty gves P( {,..., } : N (u) M) = P(N (u) > M) P(N (u) M) e M E e N(u). For ay {,..., }, E e N(u) = Y j= ˆEj `e ½{ξ(u)=} j where ξ uder P j s a cotuous-te Marov cha wth trasto rates Q = (q j), ad whch starts at ξ(0) = j. If p(j,, u) = P j(ξ(u) = ), oe gets for u t 0 > 0 ad N wth K E e N(u) = e P j= j log( ( /e)p(j,,u)) e ( /e)p(t 0) e K( /e)p(t 0) wth p(t 0) = f u t0,j p(j,, u). Note that sce, for ay, j, p(j,, u) > 0 for ay u > 0 ad p(j,, u) π > 0 as u +, oe has that p(t 0) > 0.
Therefore, for u t 0 ad wth K, tegratg o the law of N 0 µ,0(t) gves P(N (u) > N 0 µ,0(t) ) ε(t,k, t 0) wth ε(t,k, t 0) = e µ t(e ) K( /e)p(t 0). I partcular, for t t 0, sup E ( N(t) ) N : K λ t µ (t t 0)( ε(t,k, t 0)). Sce by assupto λ < µ, t s ot dffcult to choose costats t,t 0 ad K such that the rght had sde s strctly egatve (for stace, t =, t 0 sall eough ad K large eough), whch gves the result. 4..2 Proof of Theore 4.2 Itutvely, t s clear that the load-depedet RLS strategy perfors better tha the load-oblvous RLO polcy, sce t sees harder uder RLS to see a epty server. Ths sple observato shows that the uber of epty servers should be part of a Lyapuov fucto, ad deed ths leads us to defe the fucto f : N R + by:, f() = ax(ǫ, ) = + ε 0() wth 0() =½{ =0} + +½{ =0} the uber of epty servers state. I order for f to be a Lyapuov fucto, the costat 0 < ǫ < has to satsfy ǫ µ < (µ λ ) γ for soe γ > 0. Let K 0() (resp. K ()) be the set of servers that are epty (resp. have a sgle clet). Deote by 0() ad () the respectve cardaltes of these sets. Let us copute the average drft f() of the Marov process N(t) uder RLS strategy. We have: λ f() = + ǫ / K 0 () K () µ µ K 0 () «λ Y (), where Y () s the rate state at whch epty servers are fed by gratg clets. If 0() = 0, there s o epty servers state ad partcular Y () = 0. We have: f() = (λ µ ) + ǫ µ < γ, because of our choce of ǫ. K () If 0() > 0, there s at least oe epty server state. Defe p() = ax. Cosderg gratos of the p() clets fro (oe of) the server(s) wth axu sze to oe of the epty servers, we obta: Y () β p(), whch esures that f() < γ whe p() s large eough, say greater tha K. We coclude the proof by cosderg the drft outsde the set F = { : f() < (K + ǫ)}. Frst rear that F s fte. The, whe / F, p() K. We deduce that for all / F: f() < γ. The postve recurrece follows. 4.2 Approxate perforace estates The syste behavor statoary rege uder RLO ad RLS strateges s extreely dffcult to aalyze. For exaple, (N(t), t 0) s ufortuately ot reversble uder these strateges. To obta estates of the steady state dstrbuto ad clet sojour tes, we use large-syste asyptotcs,.e., we let grow large. Recetly, large-syste asyptotcs have bee successfully appled ay cotext coucato systes. They have bee used for exaple to uderstad load balacg ssues such as those arsg the superaret odel [3,7]. I the rest of the secto, we deote by N () (t) the vector represetg the ubers of clets at te t at each server a syste wth servers uder ether RLO or RLS algorth. I what follows, we cosder hoogeeous systes where λ = λ ad µ = for all. Ths restrcto splfes the otato ad results, but s ot essetal. We dscuss at the ed of ths subsecto how to deal wth heterogeous systes. We also assue that the uber of clets assocated to a gve server s bouded by a (possbly very large) costat B. Aga ths assupto s ot crucal, ad ca be relaxed at the expese of a ore volved aalyss. 4.2. RLO algorth We frst cosder RLO algorths. We assue that a clet jups fro oe server to aother at the stats of a Posso process of testy β, ad that the ext server s chose uforly at rado. The aalyss ca be exteded to ay rado wal (see 4.2.3). We represet the syste state at te t by () (t) the proporto of servers wth exactly clets at te t. We also defe S () (t) = Pl () l (t). Let us copute the average chage the syste state a sall terval of te of durato dt, ad ore specfcally the chage () creases () decreases () the chage () (). Arrvals occur at rate λ: A arrval f t occurs at servers wth clets, ad f t occurs at servers wth clets. Hece due to exogeous arrvals s dtλ( () ). Departures ca be aalyzed slarly. Let us ow copute the chage due to clet gratos. Clets gratg to server wth (resp. ) clets crease (resp. decrease) (). I addto, clets gratg fro servers wth (resp. + ) clets decrease (resp. crease) (). The average chage () due to clet gratos s thus: dtβ(( () () ) P j j() j () + ( + ) () + ). I durg dt s: suary, the average chage ()» dt λ( () () ) ( () +βˆ( () () ) j j () j () + ) () + ( + ) () +. There s o explct depedece, ad hece we expect the dyacs of () (t) to be close to those of a deterstc soluto x of the followg sets of dfferetal equatos: for all {0,..., B}, ẋ = λ(x x ) (x x + ) +βˆ(x x ) j jx j x + ( + )x +, ()
wth the coveto that x = 0 = x B+. We ay wrte slar dfferetal equatos for the evoluto of S (). We obta: for all = 0,..., B, ṡ = (λ + β j s j)(s s ) ( + β)(s s + ), (2) wth the coveto that s = 0 = s B+. Next we forally justfy the above aalyss ad show that () gves a estate of syste behavor that becoes exact whe. Traset rege. The ext theore states that the approxato s exact over fte te-horzos, ad s a drect applcato of Kurtz s theore, see Chapter [9]. Theore 4.3. Assue that l () (0) = x(0) alost surely. Fx t > 0. We have: alost surely, l sup () (u) x(u) = 0, (3) u t where x( ) s the uque soluto of () wth tal codto x(0). Proof. Frst, oe ca easly represet the faly of processes ( () (t), t 0) as a faly of desty depedet populato processes as for exaple defed [9]. The, defe F : R B+3 R B+3 by: for all x R B+3, F (x) = 0 = F B+(x) ad, for all = 0,..., B, F (x) = x (λ + β j jx j) x (λ + β + ) + x +. Now () wrtes ẋ = F(x). F s Lpschtz o T = {x R B+3 + : x = 0 = x P B B+, =0 x = }. As a cosequece, the codtos of the theore stated [9] p 456 are et, ad we deduce the expected result. Statoary rege. The above theore holds for fte te-horzos oly. It does ot say aythg about the logter behavor of the syste ad partcular for exaple about the average statoary clet sojour te. To crcuvet ths dffculty we ay use the advaced fraewor foralzed by Szta [22] ad further developed [3], ad ore recetly [6]. Due to space ltatos, we sp all detals. We vte the reader ether to verfy that results [6] apply here or to follow step by step the arguets [3] to prove the covergece of the steady-state behavor of fte systes towards the equlbru pot of dyacal syste () whe. More precsely, deote by eq () the statoary eprcal dstrbuto of the syste wth servers (such dstrbuto exsts because (N () (t), t 0) s a rreducble fte-state Marov process, ad thus postve recurret). Theore 4.4. Assue that fro ay tal codto T, the soluto of () coverges to a uque equlbru pot ξ. The () eq coverges to ξ whe. Fro the prevous theore, we ow that a syste of servers, the proporto of servers hadlg clets the statoary rege gets close to ξ as grows large. We ay also approxate the average uber of clets the syste by P ξ ad deduce a estate of the average sojour te usg Lttle s forula. It reas to show that the syste of dfferetal equatos () coverges to a uque equlbru pot ξ, ad to characterze ξ. Let ξ be a fxed pot of (), the we easly see that: for all =,..., B, ξ = ξ 0 (λ + βy) Q j= ( + βj), where y = P j jξj. ξ0 s obtaed so that ξ s a probablty easure. Fally, y ust solve:» y + B (λ + βy) Q j= ( + βj) = B (λ + βy) Q (4) j= ( + βj). Oe ca chec that f λ <, (4) deed has a uque postve soluto y: f z = λ + βy, z ust solve g(z) = 0 wth: g(z) = (z λ)[ + B z B Q j= ( + βj)] z Q j= ( + βj). The result follows fro g(λ) < 0 ad g (z) 0 for all z 0. I suary the uque equlbru pot of () s ξ. Theore 4.5. Fro ay tal codto x(0) T, f λ < µ, the syste of dfferetal equatos () coverges to the uque equlbru pot ξ. Proof. The syste ejoys the followg portat ootocty property. Cosder two tal codtos x(0) ad x (0) such that x(0) st x (0), the f x ad x are the solutos of () wth respectve tal codtos x(0) ad x (0), we have at ay te t 0, x(t) st x (t). The proof of ths property s based o a probablstc terpretato of the dyacal syste () as the Kologorov equatos of a collecto of brth-death processes of brth rate λ+β P j jxj ad death rates (+β) state. The dea s that for ay s 0, x(s) st x (s) ples that P j jxj(s) P j jx j(s), so the brth rate at te s for x s saller tha that for x, ad by a stadard couplg arguet, we deduce that just after te s, we stll have x(s+) st x (s+). We ay further deduce that ths orderg reas vald over te. Deote by x E (resp. x F ) the soluto of () whe the syste s tally epty x E (0) = (, 0,..., 0) (resp. full x F (0) = (0,..., 0,)). A drect cosequece of the above ootocty property s that x E (t) (resp. x F (t)) s stochastcally creasg (resp. decreasg) over te. For exaple, for all h, t 0, x E (t + h) st x E (t). Ths ples that both x E (t) ad x F (t) coverge to ξ whe t (sce the equlbru pot s uque). We deduce that such covergece also holds startg fro ay tal codto x(0), sce aga due to the ootocty property x E (t) st x(t) st x F (t) for all t. 4.2.2 RLS algorth The large-syste approxato ethod developed above apples to RLS algorths. We ca slarly derve a deterstc approxato for the evoluto of the syste eprcal easure (). Whe, ths evoluto s st deotes the usual strog stochastc order,.e., f x, y are probablty easures o {0,..., }, x st y ff for all j, P j =0 x P j =0 y.
characterzed by: for all = 0,..., B, ẋ = λ(x x ) (x x + )» + β x jx j x jx j x j + j 2 j +2 x j + ( + )x + j x j, (5) wth by coveto x 2 = x = x B+ = x B+2 = 0. Aalyzg the dyacal syste (5) s ot straghtforward ad deserves a full study, whch we sp here due to space ltatos. I all uercal experets preseted below, we verfed the covergece of (5) to a uque equlbru pot. 4.2.3 Exteso to heterogeous systes ad arbtrary rado wals (for RLO algorth) The above asyptotc aalyss has bee splfed by cosderg hoogeous systes ad ufor rado wals (for RLO) oly. However, the case of RLS algorth, t ca be easly exteded to the case of heterogeous systes, where the arrval rates ad server speeds are ot detcal. To do so, we ay classfy server accordg to ther arrval rate ad speed - servers of the sae class have sae arrval rate ad speed. The, we ca derve a set of dfferetal equatos, slar to () or (5), approxatg the evoluto of the proporto of servers of a gve class ad hadlg a gve uber of clets. We obta a dyacal syste whose varables x v, represet the proporto of servers of class v havg clets. I the case of RLO algorth, the aalyss ay also be exteded to arbtrary rado wals; t suffces to clude to the server class the rates at whch clets jup towards other servers. For exaple, servers of class v have the sae arrval rate ad speed, ad the rate at whch a clet at oe of class-v servers jups to a server of class v depeds o v ad v oly. I [6], the authors preset such ult-class asyptotc aalyss detals. 4.3 Nuercal experets We ow llustrate the results derved ths secto va sple uercal experets. To evaluate the relatve perforace of RLO ad RLS algorths, we cosder frst a hoogeous syste (for all, λ = λ, µ = ), ad the a extree heterogeous syste where all clets arrve at the sae server (λ = λ, ad for all 2, λ = 0). The syste perforace s expressed ters of the average clet throughput, defed as the verse of the average sojour te. Fgure gves the average clet throughput as a fucto of λ hoogeous systes. We copare the results obtaed through the large-syste asyptotcs = ad those obtaed for = 0 servers. Note that the asyptotcs results are pretty accurate eve for sall systes. Actually at a load of 0.8, the relatve error ade our approxatos of the average throughput uder RLO ad RLS algorths s less tha 4% whe = 5, ad becoes less tha 0.5% for = 20. Note that RLO ad RLS are both stable f ad oly f λ <. Surprsgly the perforace proveet acheved by the load-depedet RLS algorth over that obtaed uder the load-oblvous RLO algorth s ot that sgfcat, typcally less tha 20%. Fgure 2 provdes the perforace heterogeous systes wth = 5 ad = 0. We provde sulato results oly, although, as explaed above, we could have obtaed aalytc asyptotc results. Aga as expected, eve f all clets arrve at the sae server, RLS ad RLO stablze the syste wheever possble (whe λ < ). The dfferece betwee the throughput acheved by RLS ad RLO s qute sall rrespectve of the uber of servers cosdered. Hece t sees that pleetg a load-depedet resaplg ad grato algorth ay ot sgfcatly prove the perforace. Mea Throughput 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 Load =0 - RLS =0 - RLO >> - RLS >> - RLO Fgure : Mea throughput uder RLS ad RLO hoogeous systes as a fucto of the load λ. β = 0.5. Mea Throughput Mea Throughput 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Load =0 - RLS =0 - RLO 0. 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Load =5 - RLS =5 - RLO Fgure 2: Mea throughput uder RLS ad RLO heterogeous systes as a fucto of the load λ. β = 0.5.
5. RELATED WORK There have bee ay studes o dstrbuted, selfsh load balacg algorths ad routg gaes closed systes, see e.g. [6] ad refereces there. Refer to [8] for a qute exhaustve survey. Much of the wor ths area has cocetrated o fdg the fastest sequece of oves that would balace the syste, also called Nashfcato []. Oe class of algorths s the eleetary step syste, frst descrbed [9] whch a sequece of best respose oves are perfored by the clets. Of course ths requres that the clets ow the status of all the other servers. I [4, 5] the authors study closed systes wth lted forato about the servers status. They cosder a sychroous syste where at each step, each server saples a ew server radoly ad f the load of the sapled server s saller, the a clet oves wth probablty (N c N )/N c, where N c s the load o the curret server ad N s the load of the sapled server. It s show that the expected te to balace the syste s O(log log + 4 ). A odfcato of ths load balacg algorth s studed [5], ad t s show that the expected te to balace the syste s O(log + log ). I [2], the author cosders clets dyacs detcal to those cosdered ths paper ad uses the potetal fucto troduced [0] to quatfy the te to acheve syste balace. It s show that the expected te to reach a balace scales at ost as O( 2 ). We provde sgfcat proveets o ths boud. I ope systes, the clet oves teract a coplcated aer wth the clet arrval ad departure processes. There s very lttle wor tryg to uderstad ths teracto. Noe of the exstg wor deals wth a syste slar to that studed here. For stace, [2] aalyzes the teracto a gae-theoretcal fraewor, where arrvals are adverseral, ad where a cetral cotroller oves clets wth the a of stablzg the syste. The perforace of the classcal wor stealg load-balacg schee has also bee studed, see e.g. [3] ad refereces there. Of course there s a abudat lterature o the perforace of classcal load-balacg schees ope systes where clets are assged to a gve server for the etre durato of ther servce, see e.g. the aalyss of the superaret odel [3,7]. To our owledge, the preset paper provdes the frst aalyss of atural dstrbuted resaplg ad grato strateges ope systes. 6. CONCLUSION I ths paper, we have aalyzed the perforace of dstrbuted load balacg schees where clets depedetly decde to resaple ad chage server to prove ther servce rate. We cosdered two atural rado resaplg ad grato strateges: A load oblvous strategy RLO where clets radoly ove fro oe server to aother wthout accoutg for the actual server loads, ad a load-depedet selfsh strategy RLS where clets radoly resaple servers ad grate oly f ther rate s proved. I closed systes where the populato of clets s fxed, we have provded a ew tght boud o the te to balace server loads uder RLS strategy. Ths te ca be terpreted as the te to reach a Nash Equlbru ths selfsh routg gae. Our boud cosderably proves the bouds avalable the lterature. But t holds oly the case of hoogeous systes where servers have detcal servce rates. It sees challegg ad terestg to fgure out how to apply our ethodology to obta bouds o the te to balace the syste the case of heterogeous systes. It ght also be terestg to vestgate the te t taes to balace the syste scearos where clet gratos are lted, the sese that fro a gve server, clets ca grate to a restrcted subset of servers (as for exaple specfed va a graph). I ope systes where clets arrve at the varous servers at dfferet rates, we provded a frst aalyss of the syste dyacs. These dyacs are coplcated as the clet arrval ad departure processes teract wth the clet grato processes. We have show that both RLO ad RLS load balacg strateges are able to stablze the syste wheever ths s at all possble. It ay appear soehow surprsg that a copletely dstrbuted ad load-oblvous algorth such as RLO ca acheve axu stablty. Usg large-syste asyptotcs, we also provded approxate estates of the ea clet sojour te. The results show that aga, surprsgly, the load-oblvous RLO strategy does ot yeld sgfcat perforace losses copared to the load-depedet RLS strategy. These fdgs are vald for expoetal servce requreets, ad t would be terestg to ow whether they rea vald for other servce requreet statstcs. A terestg exteso of the preset wor (especally relevat whe cosderg spectru sharg ssues) s to aalyze the case where clets ay use resources fro several servers sultaeously. There are soe prelary results ths drecto [5], but ether the te to reach equlbru or the populato dyacs are studed. 7. REFERENCES [] IEEE Coferece o Dyac Spectru Access. http://www.eee-dyspa.org/. [2] D. Ashelevch ad J. Kleberg. Stablty of load balacg algorths dyac adversaral systes. SIAM Joural o Coputg, 37(5):656 673, 2008. [3] P. Berebr, T. 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