Load Balancing Control for Parallel Systems

Transcription

1 Proc IEEE Med Symposum o New drectos Cotrol ad Automato, Chaa (Grèce),994, pp66-73 Load Balacg Cotrol for Parallel Systems Jea-Claude Heet LAAS-CNRS, 7 aveue du Coloel Roche, 3077 Toulouse, Frace E-mal heet@laasfr, Phoe : (33) , Fax : (33) Abstract The addressed problem s to costruct ad to mplemet a load balacg polcy a Dscrete Evet System wth multple routg choces The costructo scheme determes optmal steady-state routg ratos, whch are ext trasformed to valuatos o the arcs of a sub- Petr et used to geerate the real-tme loadg decsos Such a mplemetato scheme performs as the optmal statc polcy the determstc cotext, ad t allows flexblty ad reactvty a perturbed or a fully radom cotext Keywords Dscrete Evet Systems, Queueg Networks, Petr Nets, Load Balacg, Beroull splttg INTRODUCTION Cotrol of Dscrete Evet Systems s a challegg doma both for theory ad applcatos Oe major ssue the desg of cotrol polces s to coclate effcecy, robustess ad flexblty I loadg ad schedulg problems, the determstc approach to optmal cotrol problems ofte fal because of the curse of dmesoalty combatoral problems The stochastc approach allows the costructo of smpler models, but uder some drastc assumptos, maly used to ft to the framework of Markov Decso Processes or Queueg Networks theory Furthermore, very few results are avalable for the desg of optmal dyamc cotrol polces The problem of optmally routg customers s basc for load balacg problems arsg may Dscrete Evet Systems such as computer systems, maufacturg plats ad telecommucato etworks Dyamc load balacg polces use the formato of the curret state of the system at each customer arrval date I partcular, the optmal routg problem for parallel M/M/ queues has bee show to admt a optmal soluto of the swtch-over type [] for dscouted cost crtera ad, uder some ergodcty codtos, for average cost crtera But the determato of the swtchg surfaces, by algorthms such as polcy terato, s dffcult, ad eve geerally utractable the case of fte-dmesoal state-spaces O the cotrary, statc load balacg polces are ofte much easer to compute Ad they ca be very smply mplemeted, because they do ot use ay real-tme observato of the system state They ca be dvded to two groups: the determstc oes, whch assg customers to queues a pre-determed perodcal order, ad the stochastc oes, whch route customers to queues acoordg to a Beroull splttg radom process wth pre-determed probabltes Theoretcal ad practcal comparsos [7] of these routg schemes smple cases have cofrmed the superorty of dyamc load balacg polces over statc load balacg polces ad of determstc statc load balacg polces over Beroull load balacg polces However, t s mportat to ote that, o the average, decsos take by statc ad dyamc optmal polces are the same: over a log tme horzo they sed the same average percetages of customers o each route Eve the smple case of parallel M/M/ queues wth dfferet mea servce rates, the optmal dyamc loadg polcy may be extremely dffcult to compute, ad the depedece of the soluto wth respect to system parameters has ot yet bee explcted The approach proposed ths paper cossts of three stages : - Resoluto of a statc optmzato problem to determe the optmal average routg ratos Several algorthms have bee costructed for solvg ths problem dfferet frameworks ad uder dfferet bodes of assumptos o release dates ad servce tmes [0], [4], [9] Ths paper gves explct expressos of the optmal steady-state loadg parameters for a system of parallel queues wth dfferet servce rates Average routg parameters are computed uder the assumptos of a Posso put, of expoetally dstrbuted servce tmes, ad uder a Beroull splttg of the put flow The frst step of the method determes whch servers should be used ad whch oes should ot be used by the polcy The, the optmal routg parameters are obtaed as the optmal soluto of a ucostraed

2 Proc IEEE Med Symposum o New drectos Cotrol ad Automato, Chaa (Grèce),994, pp66-73 problem - Costructo of a cyclc scheme achevg a determstc mplemetato of the optmal routg ratos The proposed method s a exteso of a orgal method proposed by H Ohl et al [8] I ths respect, the cotrbuto of ths paper s maly the techque for computg the arc valuatos of the cotrol subetwork mplemetg the optmal routg ratos - Itroducto of sequecg flexblty by a crease of the umber of tokes the cotrol subetwork [8] If the umber of extra tokes s large eough, the modfed cotrol scheme ca compesate for a temporary uavalablty of some servers The frg of some extra trastos the cases of falures ad of local overloads s a complemetary meas for short-crcutg uavalable routes 2 A STEADY-STATE FLOW DISPATCH- ING POLICY Cosder a ope queueg etwork whch admts a exact, geeralzed, or approxmate product form Its radom put flow s supposed to admt a Posso dstrbuto Upo ts arrval, each customer (of a gve class) s routed to the queue of server wth probablty α ( 0 α ) Ths Beroull splttg process does ot chage the Posso property of the put, ad the cotrolled etwork stll admts a product-form The optmal Beroull parameters are the optmal steady-state assgmet probabltes for the varous possble routgsi geeral, they ca be computed by classcal teratve algorthms [4], [5], [9] 2 A smple routg problem The ma result of ths secto s to gve explct expressos of the optmal routg parameters for a system of parallel expoetal servers wth fte queues ad dfferet mea servce rates (µ > 0; =,, ), as represeted o Fg α α α µ µ µ Fgure : A system of parallel queues The radom put flow s supposed to be Posso wth mea arrval rate > 0 Ths ope etwork s of the Jackso type [] The steady state probablty dstrbuto of customers the system s gve by : P (y,, y ) = p (y ) p (y ) () wth, for =,,, p (y ) = ( ρ )ρ y ad ρ = α µ (2) If ρ < for =,,, the average value of the tme spet the system (or mea respose tme) s : E(T ) = α µ α (3) By Lttle s formula [6], mmzato of ths performace dex s equvalet to mmzg the total average umber of jobs the etwork Uder a uform weghtg of the vetory costs, such a mmzato also correspods to the mmal average total vetory cost Wthout loss of geeralty, the servers ca be ordered the decreasg order of ther mea servce rate: µ µ (4) The costrats of the problem are related to the followg requremets: () The cotrol polcy has to be feasble Ths codto s satsfed uder the followg costrats: 0 α (,, ) ; (5) α = (6) (2) Stablty of the etwork requres restrctos: < µ (7) α µ < for =,, (8) Codto (7) also plays a crucal role the study of dyamc loadg polces I that cotext, t guaratees the exstece of cotrol polces for whch the system s ergodc ad the mmal average cost s fte []

3 Proc IEEE Med Symposum o New drectos Cotrol ad Automato, Chaa (Grèce),994, pp Optmal Routg Parameters Assume that codto (7) s satsfed by the problem data The optmzato problem, deoted (CP), for crtero E(T ) takes the form: subject to: ad: m α,,α α (µ α ) (9) 0 α m(, µ ) for =,,, (0) 0 α j m(, µ ) () j= Costrats are lear ad for =,, E(T ) µ = α [µ α ] 2 > 0 (2) Therefore, wth ts set of costrats, ths problem s covex 22 The ucostraed optmalty codtos Cosder frst the case whe the optmum of the costraed problem s also the optmum of the ucostraed problem, amely, whe the global mmum of (9) satsfes (0) ad () I ths case, the optmal soluto s smply obtaed by solvg the frst order optmalty codtos of the ucostraed problem, that s, for =,, : de(t ) dα = µ [µ α ] 2 µ [µ ( j= α j)] 2 = 0 (3) If costrats (0) ad () are satsfed at the ucostraed optmum of (9), the frst order optmalty codtos yeld, for =,, : µ α µ ( j= α j) = µ (4) µ The, for ay (,, ) ad for ay j (,, ), (µ j α j ) = µ j (µ α ) µ (5) Summg over j the 2 terms of ths equato yelds: j= µ j = (µ α ) µ j= µj (6) Ad therefore, for (,, ), α = (µ τ µ ), wth τ = ( j= µ j ) j= µj (7) By costructo, ths set of parameters satsfes: α = ( µ τ µ ) = The global mmum of (9) s feasble (ad therefore optmal) for the costraed problem f ad oly f: 0 µ τ µ m(µ, ) for =, (8) Stablty codto (7) mples that τ s postve The, codto (8) ca be replaced by: 0 µ τ µ for =, (9) ad the left part of these equaltes becomes equvalet to: µ τ 2 for =, (20) 222 Solvg the costraed problem If the left part of equalty (9) s volated for some 0 ; 0 < 0, t s also volated for ay ; 0 The, a restrcted choce problem ca be formulated, for whch α = 0 for = 0,, To show the relevace of the problem, defe the optmalty parameter assocated to routgs restrcted to the m frst routes: τ m = ( m µ ) m µ (2) Uder the coveto µ + = 0, τ m ca be defed for m =, + Its evoluto comples wth the followg lemmas: Lemma Evoluto of τ m for m =,, follows ths rule: If µ m+ < τ m µm+ the, τ m+ < τ m If µ m+ = τ m µm+ the, τ m+ = τ m If µ m+ > τ m µm+ the, τ m+ > τ m Proof By defto of τ m for m =, (2), m+ τ m µ τ m µm+ = m+ µ µ m+ Ad the rule preseted Lemma smply expresses the followg relato: µ m+ τ m µm+ = (τ m+ τ m ) m+ µ (22)

4 Proc IEEE Med Symposum o New drectos Cotrol ad Automato, Chaa (Grèce),994, pp66-73 Lemma 2 The evoluto of the optmalty parameter, τ m s creasg wth m for m m It takes ts maxmal value for m ( m ), whch s the smallest dex such that: { m µ >, µ m + τm 2 (23) The, the value of τ m mootoously decreases wth m for m < m + Proof No-emptess of the set of dces defed by (23): For ay set of rate parameters (, µ,, µ ), stablty codto (7) mples τ > 0 The, uder coveto µ + = 0, relatos (23) are satsfed for m = The set of dces satsfyg (23) s ot empty ad fte It has a mmal elemet: m Evoluto of τ m : The frst asserto of Lemma 2 s operatve f m = If m 2, cosder the values of dex m the rage (, m ) - If m µ, the, τ m 0 Ad µ m+ > 0 mples µ m+ > τ m µm+ The, from the thrd asserto of Lemma,, τ m+ > τ m - If m µ >, the, τ m > 0 For m < m, osatsfacto of (23) requres µ m+ > τ 2 m, equvalet (for τ m > 0) to µ m+ > τ m µm+ mplyg τ m+ > τ m - I partcular, τ m < τ m Ad, from the defto of τ m (2) at m ad m, µ m τ m µm = (τ m τ m ) Therefore, m µ (24) µ m τ m µm > 0 (25) Now, for m < m +, m µ m µ > The, τ m > 0 From (23), µ m + < τ m µm + Hece, from Lemma, 0 < τ m + < τ m Replace ow m by m + relato (24) to obta: µ m + < τ m + µm + Ad thus for m <, µ m+2 µ m+ < τ 2 m + From Lemma, t mples 0 < τ m +2 < τ m + Repetto of ths argumet to successve values of m up to + shows that the value of τ m mootoously decreases wth m for m > m +, ad fally that the maxmal value of τ m s obtaed for m The costraed problem ca ow be solved usg the followg proposto: Proposto Uder the feasblty codto < µ, cosder the smallest dex, m, wth m satsfyg codtos (23) The, the optmal choce of routg parameters satsfes: { αj = 0 for m < j, α = (µ τ m µ ) for =,, m (26) Proof Feasblty of the proposed polcy: The set of routg parameters defed Proposto satsfes: m α = α = m ( m µ τ m µ ) = If m = satsfes (23), relatos (26) defe the set of feasble routg parameters : (α =, α = 0 for = 2, ) If m 2, sce µ µ m for =, m, relato (25) wth τ m > 0, mples: µ τ m µ > 0 for =,, m (27) The, from (26), α 0 for =, m Furthermore, the rght part of equalty () (α µ ) s satsfed for =, m, from τ m > 0 Therefore, the parameters of Proposto (26), defe a feasble soluto to optmzato problem (CP) Optmalty amog polces satsfyg (23) ad (26): Suppose the exstece of a dex k; m < k also satsfyg codtos (23) The selected routg parameters assocated to the k routes problem s: α j = 0 for k < j (28) α = (µ τ m µ ), for =,, k (29) Deote J k = α (µ α ) wth α defed by (28), (29) Replace α by ts expresso to obta: k J k = µ k τ 2 k

5 Proc IEEE Med Symposum o New drectos Cotrol ad Automato, Chaa (Grèce),994, pp66-73 If both m ad k satsfy relato (23), use Lemma 2 to obta τ k τ m ad the majorato: J k J m + (k m )µ k τm 2 (k m ) The, µ k µ m + τm 2 mples: J k J m The choce of the smallest value of m for whch relatos (23) are satsfed s therefore optmal the class of polces cosdered ths paragraph Optmalty relatvely to ay other polcy By costructo, the polcy s optmal f m = If m <, the proposed soluto s the best amog those satsfyg α j = 0 for m < j Uder the covexty property of the problem, t ow suffces to show that the crtero caot decrease uder ay ftesmal admssble move wth δα j > 0 for ay j; m < j, startg from soluto (26) δe(t ) = µ δα j + µ j [µ ] 2 δα (30) m uder the admssblty costrat: Usg relato form: Codtos (23) mply: Hece, δα j + δα = 0 (3) m µ [µ ] 2 = τ 2 m, relato (30) takes the δe(t ) = µ j δα j + τ 2 m δα (32) m µ j τ 2 m for j = m +,, (33) δe(t ) = ( µ j τ 2 m )δα j > 0 (34) Note that partcular, f µ >, τ = µ µ The, the steady state polcy characterzed by: α =, α = 0 for = 2,, (35) s optmal f ad oly f µ 2 τ 2 3 TOWARDS A REAL-TIME IMPLEMEN- TATION OF ROUTING PARAMETERS Oce optmal average routg ratos have bee computed, they ca be mplemeted ether drectly through a stochastc or a determstc state-depedet loadg polcy, or ca be used as referece values for buldg a dyamc loadg polcy Implemetato oly uses the results of the statc optmzato scheme Ad thus, at ths stage, the techque used for computg the average routg ratos does ot actually matter So, partcular, the techque preseted ths secto ca be appled to systems much more complex tha the smple oe aalyzed secto 2 I terms of mea respose tme whe there s o falure, superorty of cyclcal dyamc load balacg polcy versus statc load balacg polcy has bee show by several authors [7] I the partcular case of Fg wth all servce rates equal, superorty of the Roud- Rob rule over the Beroull splttg techque was theoretcally establshed Ephremdes et al [3] Furthermore, cyclcal sequecg s a good termedate stage the desg of a flexble loadg devce whch sequecg flexblty ad routg flexblty ca be troduced [8] 3 Costructo of a cyclcal routg devce Routg alteratves are ow represeted as a freechoce place, labelled 0, a Petr et represetg the system O Fg2, a frg of trasto correspods to a release o route 0 Fgure 2: A free-choce Petr et represetato A cyclcal frg sequece for routes, s obtaed by troducg places for decso tokes ad valuated arcs betwee these places ad the route put trastos To evaluate the average steady state routg

6 Proc IEEE Med Symposum o New drectos Cotrol ad Automato, Chaa (Grèce),994, pp66-73 ratos, a fte feedg of tasks at place 0 ca be assumed The, place 0 ca be suppressed, ad oly the dstrbuto of tokes determes the system evoluto The logcs of such a cyclcal loadg devce are represeted o Fg3 p p - p - p p Fgure 3: A Petr et represetato of the loadg mechasm The cyclcal Petr et characterzg the loadg devce s represeted by the followg cdece matrx : p 0 p C = p p 2 0 (36) 0 p The varat T-semflot : u R such that Cu = 0 s gve by : j u = p j k= [ j k p (37) j] If the Petr et of Fg3 s lve, the varat T-semflot defed by the frg sequece u characterzes the cyclcal statoary behavour of the Petr et [2] To obta the desred loadg ratos α, α, t suffces to determe valuatos p,, p whch satsfy the set of equatos : u = α for,, or equvaletly, for,, p j = α P wth P = j 0 p k=[ j k p p p p j ] ad p > 0 (38) From expresso (37), t s clear that parameters p ca be multpled by ay postve costat wthout chagg the routg ratos Furthermore, for the set of parameters (p,, p ) to be used as a set of valuatos o the arcs of the cotrol Petr et, t has to be approxmated by a set of postve tegers Thus, f system (38) ca be solved for a set of postve real parameters (p,, p ), the order of magtude of parameters p has to be selected so as to acheve a trade-off betwee - a suffcet precso spte of the roudg to the earest tegers - a reasoable umber of tokes the Petr et Such a trade-off may deped o the cosdered applcato If precso s ot essetal, the smallest valuato (p uder a decreasg orderg of the α ) ca be set equal to If routg ratos have to be more precsely met, the value of p should be creased, as the Example of the followg secto, where the value of p s set equal to 0 To solve the system of equatos (38) terms of real postve parameters, take the atural logarthm of each term of the equatos to obta the followg set of lear equaltes (wth ucostraed real varables ad P arbtrarly set to ) : Ay = b (39) 0 0 wth A = 0 0 y T = [y,, y ] wth y = Log(p ) b T = [y,, y ] wth b = Log(α ) Matrx A s square ad regular for 2 ad ts verse s explctely gve by : ( 2) A = ( 2) ( 2) Soluto of (39) s y = A b, ad the correspodg vector of real valuatos, p = [p,, p ] T has postve compoets It s obtaed by : p = e y (,, ) (40) For a ormalzed value of p equal to π ( π = or π = 0 for stace), the vector of teger valuatos π = [π,, π ] s obtaed as : π = roud{ π p p} (4) where roud() rouds a postve real umber to the closest o-egatve teger Now, the codto to be satsfed to obta the desred routg ratos s lveess of the Petr et Assumg that

7 Proc IEEE Med Symposum o New drectos Cotrol ad Automato, Chaa (Grèce),994, pp66-73 there s o blockg o ay of the routes, a ecessary ad suffcet codto for lveess ca be obtaed as a exteso to a result by Ohl et al [8] to the case of routes Vector x = [,, ] T s a varat P- semflot of the cotrol Petr et : x T C = 0 Therefore, f m deotes the umber of tokes place, the total umber of tokes the cotrol Petr et, N = s a costat alog the system evoluto - f N π, all the trastos may be smultaeously dsabled, for some partcular markgs, - f N > π, there s always at least oe trasto eabled, ad sce the cotrol Petr et s a elemetary cycle, all the trastos are lve Therefore, the lveess codto s as follows : The cotrol Petr et s lve f the umber of tokes the cotrol Petr et, N, satsfes : N m π + (42) Eve for N = π +, f > 2, several trastos may be smultaeously eabled, ad some decsos rema to be take Ad release flexblty creases wth N, for N > π + Ths feature s basc for avodg mmedate geeral blockg whe a falure occurs 32 Itroducg routg flexblty Uder codto (42), a crease of the umber of tokes the cotrol graph allows ew state trastos For some markgs, several trastos are smultaeously eabled, ad dfferet sequeces ca be geerated Coflcts ca the be solved usg prorty rules whch may be statc or may deped o the state of the etwork However, uder ay feasble coflct resoluto strategy, the same average routg ratos are satsfed, sce the varat T-semflot remas the same I the queueg etwork of secto 2, the capacty of queues was supposed fte ad the servers were always avalable A basc exteso of ths model cossts of troducg resource avalablty codtos o the frg of trastos,,, represeted o Fg 4 by the presece of a toke place,, The, f a falure occurs, causg a blockg o route, the avalablty codto caot be satsfed ad cotrol tokes ted to accumulate at place After some tme, all the trastos j, j get dsabled by a lack of tokes Such a accumulato of tems ca be used to detect the falure, ad the repar acto ca start The process of routg optmzato ad mplemetato may also be recomputed However, t ca also be terestg to rapdly react to such a falure by short-crcutg the uavalable route Ths ca be doe by addg ew trastos τ from ode to ode +, wth valuated arcs q, as o Fg 4, so that τ gets eabled f the umber of tokes place gets bgger tha q If the total umber of tokes, N > π +, s selected large eough to always have at least 2 cotrol places wth m p (or at least oe cotrol place wth m 2p ), N π + π + π, a possble choce of q s obtaed cosderg the smallest markg of place for whch all the routes j are possbly dsabled by a lack of tokes It s : q = N j π j + q p p τ q p p - q p τ q 0 p - p τ q p Fgure 4: Cotrol devce wth avalablty codtos ad Supervso No lveess of the supervsory cotrol cycle s ormal sce ts trastos should be eabled oly perturbed codtos ad o a o permaet bass Several dfferet techques ca be thought of to crease routg flexblty The supervsory scheme preseted here s just a tetatve adaptve verso of the proposed cotrol structure Its terest s maly to set to evdece the potetal flexblty of the valuated cyclcal routg techque 4 EXAMPLE Cosder a system of 5 parallel queues, wth expoetally dstrbuted arrval tmes ad servce tmes wth mea rates: =, µ = 08, µ 2 = 06, µ 3 = 04, p

8 Proc IEEE Med Symposum o New drectos Cotrol ad Automato, Chaa (Grèce),994, pp66-73 µ 4 = 02, µ 5 = 0 Computato of the optmalty parameter τ m for m =,, 5 gves: τ = 02236, τ 2 = 02397, τ 3 = 03476, τ 4 = 03638, τ 5 = The, from Proposto, the optmal steady-state Beroull splttg follows the dspatchg probabltes: α = 04746, α 2 = 0382, α 3 = 0699, α 4 = 00373, α 5 = 0 I steady state codtos ad the mea, server 5 should ot be used Ths property remas vald for values of less tha 5 3 But server 5 should be used wth a postve probablty for > 5 For =, uder the Beroull splttg polcy wth routg probabltes α, α 2, α 3, α 4, α 5 gve below, the mea system respose tme s gve by E B (T ) = 4 α µ α 3555 From these values, t s possble to buld a determstc statc load balacg polcy by costructg a cotrol Petr et as secto 2 Oly the frst 4 routes are cosdered The vector of valuatos π s computed by relato (4), after solvg system (39) The selected value of π s 0 It gves : π = [ ] T Implemetato of ths cotrol Petr et gves the followg vector of average a posteror routg parameters : α = [ ] T The codto for lveess of the cotrol Petr et s that the sum of the tokes places, 2, 3, 4, deoted N, satsfes : N 77 Costrats o mache avalablty ad the supervsory cotrol structure of Fg4, preseted secto 32 ca be tegrated wth, for stace, N = π + π 2 + 2π 3 + 2π 4 4 = 422, ad the vector of valuatos : 5 CONCLUSION q = [ ] T Statc load balacg polces are geerally much smpler to compute tha dyamc oes However, the cocers for flexblty ad reactvty of routg cotrol devces urges to the use of dyamc load balacg polces I ths paper, explct expressos of the optmal steady-state routg probabltes have bee gve for the case of parallel M/M/ queues wth dfferet mea servce rates I partcular, a smple test based o the computato of the optmalty parameters, τ m gves a threshold value (τm 2 ) for the mmal mea servce rate of the servers to be used by the polcy Istead of a drect mplemetato of the optmal steady-state routg parameters as Beroull splttg parameters applyg to the put flow, t s proposed to mplemet a cyclcal cotrol Petr et achevg optmal steady-state performace Some smple techques ca the be troduced to crease the atural flexblty of the cotrol devce ad ts adaptvty case of perturbatos Refereces [] FBaskett,KMChady, RRMutz, FGPalacos Ope, closed ad mxed etworks of queues wth dfferet classes of customers JACM, vol22, No2, pp , 975 [2] GWBrams Réseaux de Petr : Théore et Pratque, Masso, 983 [3] AEphremdes, PVaraya, JWalrad A smple dyamc routg problem IEEE Tras Automatc Cotrol, AC-25, No4, pp , 980 [4] YFre, YDallery, JPerrat, RDavd Optmsato du routage des pèces das u ateler flexble par des méthodes aalytques, APII vol22, 988, pp [5] JCHeet, CASPassos, KSmal Optmal routg of products a queueg etwork represetg a FMS ECC 9, Greoble, Frace, 99 [6] LKlerock Queueg Systems, vol, Wley Iterscece,976 [7] RDNelso ad TKPhlps A approxmato for the mea respose tme for the shortest queue routg wth geeral terarrval ad servce tmes Performace Evaluato, 7 (993), pp23-39 [8] HOhl, ECastela, J-CGeta State depedet release cotrol Flexble Maufacturg Systems, IEEE-SMC Cof, Le Touquet, 993 [9] K Smal, JC Heet Optmsato du routage des pèces das u ateler flexble à cotrates de capacté locales RAIRO-APII, Vol 26, No 3,pp , 992 [0] ANTataw, DTowsley Optmal Statc Load Balacg Dstrbuted Computer Systems, JACM, vol32, No2, 985, pp [] JWalrad A troducto to queueg etworks Pretce-Hall, 988