Performance Prediction of Distributed Load Balancing on Multicomputer Systems

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1 Performnce Prediction of Distributed Lod Blncing on Multicomputer Systems Ishfq Ahmd *, Arif Ghfoor+, nd Kishn Mehrotr * * School of Computer nd Informtion Science, Syrcuse University, Syrcuse, NY school of Electricl Engineering, Purdue University, West Lfyette, IN 4797 Abstrct Thispperpresents performnce evlution pproch to compre different distributed lod blncing schemes on unified bsis. This pproch is n integrtion of simultion, sttisticl nd nlyticl models, nd tkes into ccount thefundmentl system prmeters tht cn possibly ffect the pe~ormnce. We show tht ll the sender+ nitited distributed lod blncing strtegies cn be modeled by centrl server open queuing network. Furthermore, these lod blncing strtegies cn be chrcterized by only two queuing prmeters the verge execution queue length nd the probbility (ht newly rrived tsk is executed loclly or m grted to nother node. To cpture the reltwn between these queuingprmeters nd vrious system prmeters, sttisticl nlysis hs been crried out on the empiricl dt obtined through simultion. The nlyticl queuing model is then used topredict the response time of system with ny combintion of systemprmeters. Experimentl results re obtm nedfor six dl~erent lod blncing strtegies. The proposed model provides pe~ormnce results in strightforwrd mnner nd cn be beneficil to the system designers in ssessing the system under vrying conditions. L Introduction Efficient utiliztion of multicomputer system lies in its bility to efficiently prtition nd blnce the computtionl lod mong its computing nodes. With the incresing populrity of multicomputer systems, reserchers nd system designers hve been focusing on these essentil issues. If the decisions to llocte worklod tsks to processing nodes re fixed nd re tken before ctully running the problem, then lod blncing is considered sttic. For dynmic lod blncing, there me no fixed lloction decisions nd lod is blnced depending upon the time dependent stte of the system. Dynmic lod blncing hs lso been termed s dynmic lod shring [3], or lod distribution [8]. As noted in [3], ny simple dynmic lod blncing lgorithm improves the performnce of the system, nd is better thn no lod blncing. Dynmic lod blncing strtegies re chrcterized by the mnner in which informtion exchnge nd control of worklod lloction tkes plce. The control cn be centrlized [14], fully distributed [2], [3], [4], [9], [12], [15], [22] or semi distributed [1]. With fully distribtttedcontrol, the lod blncing strtegy is incorported t every node of the system in tht ech node in rhe system mkes utonomous decisions. A node decides whether the tsk submitted to it should be executed loclly or trnsported to some other node. If the tsk should be migrted, the locl node needs to know the lod sttus of other nodes. Anode for tsk migrtion cn be selected rndomly [31, [61, [221 or with some other criteri [3]. However, the ccurcy of scheduling decisions in decentrlized lgorithms, depends on the ccurcy nd mount of stte informtion [8]. Wng nd Morris [23] proposed number of reltively simple lod blncing lgorithms nd clssified them into two ctegories: source-initited nd server-initited. In source initited lgorithm, tsks enter the distributed system vi source nodes nd reprocessed by server nodes. Fox et L [5] presented lod blncing scheme by mking use of the nlogy of lod blncing to minimizing n pproprite energy function. In[15] nd [2], vrious bidding lgorithms hve been proposed, which belong to the sender-initited clss. A drfting lgorithm belongs to the server-initited clss [13]. A comprison of these two types of lgorithms [16] revels tht in spite of the fct tht the bidding lgorithm suffers tlom tsk-dumping or tsk thrshing, it performs consistently better thn the drfting lgorithm. Tsk-thrshing is phenomenon ssocited with lod blncing where lightly loded node cn become victim of tsk rrivls from other nodes [6], [12]. Lod blncing lgorithms cn lso suffer from stte woggling nother performnce decying phenomenon in which processors frequently chnge their sttus between low nd high [16]. For systems with certin interconnection topologies, distributed lod blncing schemes bsed on tsk migrtion mong nerest neighbors hve gined considerble ttention. In number of independent studies [6], [8], [1], [17], vrints of this strtegy hve been proposed nd their effectiveness hs been proven both by simultion nd implementtion observtions. KE [ 17] hs compred one version of this strtegy, known s Contcting Within Neighborhood (CWN), to the Grdient Model [11] nd hs shown tht CWN spreds the lod more quickly nd performs better. In two more studies [6], [17], the concept of lod verging mong neighbors is introduced. The dvntge of lod verging is tht ech node tries to keep its own lod equl to the verge lod mong its nettrest neighbors. Shu nd K16 [19] hve proposed nd implemented revised version of CWN known s Adptive Contrcting Within Neighborhood (ACWN), which consistently shows better response time compred to the Grdient model nd Rndom strtegy. Grunwld et l. [8] hve proposed clssifiction 1991 ACM /91/83 $1.5 83

2 for the type of informtion required to mke lod blncing decisions. Given the diversity of number of proposed strtegies nd their dependence on number of prmeters, it is difflcult to compre their effectiveness on unified bsis. One prticulr strtegy my perform well under certin combintion of prmeters, such s, system lod or system communiction rte on certin topology. The sme strtegy my be outperformed by nother strtegy due to difference in informtion collection nd scheduling overhed. In ddition, simplified ssumptions nd neglecting importnt prmeters sometimes obscures the reltive merits nd demerits of ech strtegy. This pper presents n pproch to predict nd compre the performnce of different lod blncing schemes bsed on unified bsis. Our pproch, which is n integrtion of simultion, sttistics nd nlyticl models, tkes into ccount vrious system prmeters, such s system lod, tsk migrtion time, scheduling overhed nd system topology etc., tht cn ffect the performnce. We show tht lod blncing strtegies, belonging to the sender-inititedclss, cn bemodeledbycentrl serverqueuing network. We lso show tht these strtegies cn be chrcterized by only two prmeters the verge queue length nd the probbility tht tsk is executed loclly or migrted to nother node. Through n extensive simultion, lrge number of vlues of the verge queue length nd theprobbility ssocited with tsk migrtion hve been obtined. A sttisticl nlysis hs been performed on these dt points to cpture the reltion between the queuing prmeters nd the system prmeters. The nlyticl queuing model is then used to predict the response time of system with ny set of prmeters. Six different lod blncing lgorithms hve been studied nd chrcterized. This performnce prediction pproch hs mny dvntges. First, insted of ssessing prticulr strtegy on the bsis of selected set of experiments, ny combintion of prmeters cn be used to predict the performnce. Second, ll strtegies cn be reltively compred by selecting more pproprite nd relistic prmetem. Finlly, n existing system cn be tuned, nd system design cn be evluted before it is ctully buik Theresponsetime predictedbythe model is compred with the response time produced by simultion for ll six strtegies. 2. Selected Lod Blncing Strtegies We consider fully homogeneous mukicomputer system in which processing nodes re connected with ech other through symmetric topology, tht is, ech-node is linked to the sme number of nodes. The number of links per node, clled the degree of the network, is considered s one of the system prmeters nd is denoted sl. The worklod submitted to the system is ssumed to be in the form of tsks, which re submitted to ech node with n verge rrivl rte of tsks per time-unit per node, f%e tsk rrivl process is ssumed to be Poisson. The lod blncing control is fully distributed for which ech node mkes n utonomous decision to schedule tsk by collecting the lod sttus informtion from its neighbom. A tsk is either scheduled to locl execution queue or it is migrted to one of the neighbors connected with ech communiction chnnel. The informtion nd scheduling tkes certin mount of time, which is ssumed to be exponentilly distributed with n verge of l/as time-units. Informtion is collected by hrdwre@ftwre component t ech node nd is clled CollcdorlScheduler. Since informtion interchnge nd execution of the scheduling lgorithm tkes certin mount of time, the tsks rriving during tht time wit in witing queue. For ech communiction link, communiction queue is mintined. The underlying network supports point-to-point communiction nd the communiction chnnel is modeled by server. A communiction server trnsfers tsk from one node to nother with n verge of 1/% time-units. The tsk communiction time is lso ssumed to be exponentilly distributed. Ech communiction queue is served on the FCFS bsis. At ech node, the incoming trftlc from other nodes joins the loclly generted trffic, nd both re hndled with equl priority. Ech node mintins n exeeution queue in which loclly scheduled tsks re served by CPU on the FCFS bsis. A tsk my migrte from node to node in the network before finlly being executed t some node. The execution time is lso ssumed to be exponentilly distributed with n verge of l/pe time-units. We hve nlyzed six different sender-initited lod blncing strtegies for vrying informtion collection mechnisms nd scheduling disciplines. Bsed on the informtion interchnge mechnism, these strtegies cn be further clssified into two ctegories. In the fwst ctegory, the informtion bout the lod nd the sttus of other nodes is collected t the time tsk is scheduled for execution or migrtion. The lod is expressed in terms of the length of the execution queue. This lod metric hs been widely ccepted nd experimentl results hve shown tht it ccurtely reflects the CPU lod [16]. In the second ctegory, nodes exchnge the lod informtion mong their neighbors perioditlly. Within ech ctegory, we hve considered three different scheduling policies. Ctego rv. I: Informtion Exchnpe t the Time of Tsk Schedule 9 FRndom: In this strtegy, the tsk scheduler clcultes the verge of the locl lod nd the lod of ll neighbors, If the locl lod is greter thn the verge, the tsk is sent to rndomly selected neighbor. If the locl execution queue is empty (or locl lod is less thn the verge), then the tsk is sent to the locl execution queue. FMin: In this strtegy, the tsk scheduler sends new tsk to the node tht hs the minimum lod. However, if the locl node s lod is equl to the minimum lod mong neighbors, the locl node is given priority. FAverge: In this strtegy, the tsk scheduler clcultes the verge 831

3 of ll neighbors lod nd its own lod. If the locl lod is greter thn the verge, the tsk is sent to the neighbor with the minimum lod. However, if the locl execution queue is empty or the locl lod is less thn the verge, the tsk is sent to the locl execution queue. II: Periodic Informt ion Exchntz prndom: This strtegy is similr to FRndom except tht every node sends it own lod informtion to ll its neighbors periodiclly. The time period, T. for sending messges is system prmeter.. PMin: This strtegy is similr to FMin except tht informtion exchnge is done periodiclly.. PAverge: This strtegy is similr to FAverge except tht informtion exchnge is done periodiclly. 3. The Performnce Prediction Model In this section, we describe the performnce prediction model for the distributed lod blncing strtegies described bove. I%is model isnintegrtion of simultion, nd sttisticl nd queuing models. First, we describe the queuing model nd show tht the clss of distributed lod blncing strtegies described bove cn be modeled by n open centrl server queuing model The Queuing Model As described bove, the multicotnputer system consideredhere is symmetric nd homogeneous. By symmetry, we men tht the interconnection network of the system is regulr grph with fixed number of links per node. By homogeneity we imply tht the processors of the system hve identicl processing speeds. Similrly, ll communiction chnnels nd tsk schedulers re identicl. The stedy-stte tsk deprture nd rrivl rtes t every node re the sme. As explined erlier, tskkeeps on migrting until it finds suitble node. When tsk migrtes from one node to nother, it sees sttisticlly identicl node. Therefore, the stedy-sttekhvior of nerest neighbor lod blncing cn be pproximted by the open centrl server queuing model s shown in Figure 1. The model consists of witing queue,l communiction queues nd n execution queue. This model is pproximte, since routing of tsks is dependent on the stte of execution queues. However, s described in the next section, simultion results obtined on ctul network topologiesre very close to thenlyticlresults determined from this model, which vlidte tht theproposedmodelof Figure 1 indeed represents the tsk scheduling nd migrtion process. The durtion oftsk sresidence time in thesystemconsists of two phses. In the first phse, the tsk my keep on migrting during the course of which it wits in the witing queue, gets service from the scheduler, wits in the communiction queue, nd then trnsfers to nother node. At tht point the sme cycle my strt ll over gin. Once the tsk is scheduled t the execution queue of node, the second phse strts, which includes the queuing nd service time t the CPU. In the fiist phse, the tsk cn be viewed s occupying either the tsk scheduler or one of the communiction links. The Mrkov chin shown in Figure 2 describes the behvior of the centrl server which in turn explins the tsk migrtion phenomenon before the tsk enters the execution queue. The stte of the Mrkov chin is described by (L. +1) tuple, h, kl,.. kl in which ki represents the number of tsks t the i-th queue ( O s i s L) t node. It follows [21] tht the model crt be solved by the Jcksonin network, which hs the product form solution; tht is, the joint probbility of kj tskst queuej (j =,1,..., L) is given by the product: Wok,,,k,...kL) = ~ Pl{kj) where ~j(kj) is the probbility of kj tsks tj-th queue nd is given by: ~j{kj) H (1 Qj) e? L j=o AA Witing Queue Scheduler Informtion Co ector ~ 111% Ps Communiction Queue I Migrted Plj ia Execution Queue Loclly submitted tsks 1- with rte2 Communiction P; Queue tsks Sink Figure 1: Distributed lod blncing represented by open centrl server model. 832

4 & Figure 2: Mrkov chin with the stte of the chin describing the number of tsks t ech queue of node. For thej-th component, the verge utiliztion, Q], is equl to ~j/pj. The eqution implies tht the lengths of ll queues re mutully independent in stedy stte. The bove model cn lso be solved while considering the probbilistic behvior of tsk, Suppose, fter the tsk is served by the scheduler, it goes to the i th link with probbility Pi or it enters the locl execution queue with the probbility P.. When tsk leves (enters) the witing queue, thenumberof tsks in tht queue is decresed (incresed) by one. Similrly, when tsk is served by the communiction, sttisticlly identicl tsk joins the witing queue. The verge queue length nd verge response time for the j th component is given by: respectively. The ~verge number of tsks t n&l e is the sum of the verge number of tsks t ech component node nd is given by. ~j of from which thewverge res~nse time before the tsk is scheduled in the execution queue cn be computed s [2 1]: = 1 / (p#o) I Pj / (p@j) 1- ~ / (f r#o) + ~, 1 - Pj O) Once tsk is scheduled t locl execution queue, the response time from the time it is scheduled to the time it finishes execution is given by: where EINEI is the verge execution queue length. The complete response time, therefore, is given by E[R] = E[R~] + E[RJ, Theboveeqution implies tht, forgivensystem lod, w nd Pj s, the response time yielded by lod blncing strtegy cn be clculted if the probbility, P., nd the vergeexecution queue length, HNEI is determined. h other words, P., is the probbility with which lod blncing strtegy schedules the tsks loclly. The probbility tht tsk will be migrted to nother node is simply 1 P. nd migrtion probbilities to individul chnnels t ech node re identicl. The verge execution queue length, E[NK], determines how smoothly the lod is blnced. Both prmeters, Po nd E[N~], depend on system prmeters, such s 1, p,, Ac, p,, T. ~dl In the next sections, we briefly describe the simultion methodology tht is used to obtin very lrge dt set from different test cses. We describe how we performed sttisticl nlysis on the simultion dt nd determined the sensitivity of POnd E[Nd ginst different system prmeters The Simultion Model The bove mentioned lod blncing strtegies were simulted on n Encore Multimx. The simultor ccepts the topology of the network long with 2, ps, PC, & Iengthof simultion run, ndchoiceof lodblncing strtegies nd their ssocited prmeters. The results produced by the simultor include verge response time, utiliztion of individul nodes, verge time spent in communiction, verge number of messges, throughput, verge number of migrtions mde by tsk nd their distribution, verge lengths of witing, communiction nd execution queues. In ddition to verge vlues, the vrince nd ech node s individul sttistics re lso produced The probbility, P., is then clculted by dividing the verge number of loctly 833

5 scheduled tsks by the totl number of tsks rrived, t ech node. The importnt spects of discrete-event simultion re tht it should be run for sufficiently long time nd initil trnsients should be removed before strting the ccumultion of sttistics. Moreover, the confidence intervl must be clculted fter rttming the sme experiment with multiple independent strems. All of these fetures hve been incorported in the simtdtorndllr esults re obtined with 99 % confidence intervl. Along series of simultion runs ws conducted to obtin totl of 5 dt vlues for POnd E[NE] were obtined, for ech strtegy. Three different topologies were selected, which included the ring, the hypercube nd the folded hypercttbe [7], ech consisting of 16 nodes. Ech point for one prticulr strtegy ws obtittedon ech of the topologies by fining one prmeter nd vrying the rest. In most cses, 1 ws vried from to.9 tsks per time-unit, PS ws vried from 8 to 16 tsks per time-unit nd PC ws vried Tble I: Estimtion for PO nd its sensitivity versus system prmeters. S&tegy FRndoon FMin FAverge PPndom PMin PAverge R-Squre System Prmeter ;inh Pc y Liti k -!&_ L Links Pc TV Linb Pc y Tu L& k y Tu Links Pc y Z!oefficlent Estimte t Note All estimtes of model prmeters re sttisticlly significn~ except slightly significnt ), nd not significnt (f). from 8 to 16 tsk per time-unit. The tsk execution rte, p~, ws fixed s 1 tsk per time-unit in ll cses. For strtegies tht required periodic informtion updte, the updte time, TM., period ws vried from.5 time-units to 1.5 timeumts. It is worth mentioning tht the simultor tkes into ccount the time to schedule tsk, which includes the exchnge of stte informtion nd the execution of the scheduling lgorithm itself. Most previous studies hve ignored this overhed. We hve ssumed n verge scheduling time, 1/,S, which in turn, cn be normlized with respect to the execution time, p,. In other words, when ~~ is 1tsks/ timeunit nd g~ is 1 tsk/time-unit, the verge tsk scheduling time is 1I1O of the execution time. We consider it n input prmeter which cn be observed from rel system depending upon how the informtion messge hndling nd regulr tsk migrtion is implemented Sttisticl Anlysis To chrcterize PO nd E[NJ in terms of system prmeters, such s ~, ~~, PC, TM nd system network topology, sttisticl nlyses hve keen performed. As described txwe, dt on PO ws collected for vrious vlues of the system prmeters for ech lod blncing strtegy. A regression nlysis ws then performed to obtin model tht expresses PO in terms of the system prmeters. It is observed tht the model shown in Eqution 1 works quite well for ll six strtegies. The estimtes of P nd coefflcients,~ s, re given in Tble I long with mesures tht describe how well the bove model predicts the observed P.. For instnce, in cse of FRndom, the R-Squre vlue is.9277, which implies tht the regression model is ble to compute YO of vrition observed vlues of P.. A similr regression nlysis pproch is tken to chrcterize E[N~] in terms of system prmeters. In this cse, the observed model is given by Eqution 2. This model fits extremely well, sis observedfiom its R Squzue vlues (ll R Squxe vlues re 99 %) given in Tble II. In cse of E[N~], the coefficients for A rtdpc were found insignificnt nd hence re ignored The Complete Model The complete model for performnce prediction is shown in Flgttre 3. The performnce mesure is the verge tsk response time. As described bove, the model building consisted of running lrge number of simultions, then pplying sttisticl nlysis to obtin models for PO nd EIIVE]. Using this model, thevhtes of PO nd E[N~] cn be directly computed for ny of the six lod blncing strtegies with ny combintion of system lod, communiction rte, tsk scheduling rte, lod updte period (for lod blncing strtegies belonging to ctegory II) nd network topology. We then compute the verge response time by USing the formul given in section 3.1. As explined erlier, this response time consists of hvo prts. The first prt is the verge response time before 834

6 %+IiW=$ + I&UC + I@ + #V + AT. I Po=[l+e- ( ) 1 Eqution E[N~] = exp (q +~,$inks + /%#c + J%pS + 1%# + I%J. ) Eqution (II) (I) Tble II: Estimtion for E[N~] nd its sensitivity versus system prmeters. Strtegy R-Squre Prmeter I I I Prmeter Estimte RmdOOml 31 I AS I MY FMin FAverge PRndom PMin PAverge A Links ps Tu Links A ;.116 Links A : -.95 Link -.52 tsk is scheduledin n execution queue. This is simply equl to the time the tsk is scheduled (in the execution queue of node) minus the tsk rrivl time. This response time, clled trnsient time, is completely described by Po, which indictes the tsk migrtion tendency of bd blncing strtegy. The second prt of verge response time shows how much time (queuing dely plus execution time) tsk tkes fter eventully being scheduled. This time is equl to the time the tsk finishes execution minus the time the tsk ws scheduled in the execution queue. The best trnsient response time results when strtegy s P. is neither very high nor very low. In other words, the strtegy should not hve tsk thrshing tendency nd yet it should mke tsk migrtions whenever pproprite. The second prt of the response time depends on strtegy s lod equliztion bility; tht is, smller verge execution queue length will result if the lod is eqully blnced. Both fctors, however, re dependent on ech other. For exmple, if strtegy suffers from tsk thrshing, execution queue length is not blnced nd the verge vlue of queue length increses. As n exrnple,figure4 shows the plot of POversus system lod for ll six strtegies, on 16-node hypercube. We notice tht t low lod both FMin nd PMin hve high vlues of PO, which shrply increse t high lod. This implies tht both Min strtegies schedule more tsks loclly (nd hence, mke less migrtions) but trnsfer more tsks t high lod. In contrs~ both rndom strtegies hve low vlues of P., which implies tht greter tsk migrtion tkes plce using rndom lgorithms. Figure 5 shows the vritions in E[NE] versus system lod for ll six strtegies. From this figure, we observe tht, in this cse, the vlue of E[N~] is the minimum with FAverge followed by PAverge, nd Ph4in results in the lrgest verge queue length. 4. Performnce Prediction, Evlution nd Comprison After obtining response time dt from the performnce prediction model, we compre it with the observed simultion results. Six lod blncing strtegies long with vrying vlues of A, #S, PC, T. nd different network topologies provide wide rnge of figures to mke comprison between the response time obtined with the model nd the response time obtined with simultion. However, we compre the two figures by vrying one prmeter while keeping the rest constnt. Theresuks re quite encourging, nd the difference between the two figures is found to be less thn ~ 7%. Since ll results cnnot be provided within this pper, we present only those results with noticebleimpct of ech prmeter on response time produced by the model, s well s by the simultion. First, we exmine the impct of system lod on the verge response time for ll six strtegies, shown in Figure 6 nd Figure 7. In both figures, we hve plotted the pirs of verge response time computed from the model nd the venge response time observed horn simultion. The difference in model nd simultion results is lso indicted on these figures. The tsk scheduling rte, ps, nd the tsk communiction rte p= re both 16 tsks/tim&unit. System topology is 16-node hypercube network nd lod updte period, T., is.5 time-units. In Figure 6, system lod Q is.5 (with A =.5 nd pe = 1). Figure 7 differs from Figure 6 in tht the system lod is incresed from.5 to.8. From these figures, we observe the following The difference in the response time computed from the model nd the response time observed from simultion is very smll. For most of the cses, this difference is less thn 1%. The worst cse difference is 6.52%. At low loding conditions, FAverge performs well, wheres PRndom performs the worst of ll. The difference in the performnce of FRndom nd PRndom is not significnt, implying tht for rndom lgorithms, informtion exchnge cn be done either instntneously w periodiclly with T. =.5. me difference in the performnce of FiWin nd PiWin is not significnt. Agin, this implies tht informtion updte cn be done by selecting either of the two principles. 835

7 ssimultion simultion Dt E Amdytict ~ Queuing Model )- ~ Model Building Prmeters Selection * Figure 3: The complete Performnce Prediction Model. E[NE] I I ~ Figure 4: Vritions in ProbbilityPO versus system lod for vrious lod blncing strtegies ~ Figure 5: Vritions in E[NE] length versus system lod for vrious lod blncing strtegies. This indictes tht with TM =.5, periodic updte strtegies perform s well s fresh informtion updte strtegies. In order to check the vlidity of the proposed model for vrious prmeters, we chnge PS nd PC but keep the rest f~ed. These results re shown in Figures 8 nd 9. A high system lod, eqol to.8, is selected by first considering fst communiction network nd slow tsk scheduling rte (PC = 16 ~ks/timeunits d PS = 8 tshhw-unk), nd then considering slow network nd fst tsk sehedulingrte with (PC =8 tsks/time-units nd ps = 16tsks/time-units). Agin, the model is shown to predict the verge response time, which closely mtches the response time produced by simultion, Further insights drwn from these figures re summrized below.. We note tht tsk scheduling time hs greter impct on the verge tsk response time thn the tsk communic- tion time. This is obvious becuse the verge response time with slow sehedulingrte nd high communiction rte (Figure 8) is greter thn the response time with fst scheduling rte nd slow communiction rte (Figure 9). The observtion is true for ll strtegies. Next, we show two rbitrrily chosen sets of system prmeters. In the fwst set, 16-node folded hyperculx with 5 links per node t reltively low system lod (.6) is selected. Thetskcommuniction rte ndthetsk scheduling rte re both 12 tsks/time-unit nd T= is equl to 1.5 timeunit, which is reltively lrge. The results for this combintion of prmeters re shown in Figure 1 nd re summrized below. The difference in the response time for the model nd simultion is gin very smtl. The periodic updte strtegies, PMin nd PAverge, re outperformed by FMin nd FAverge becuse of the lrger vlue of Tw. 836

8 Comprison of response times predicted by the model nd simultion for vrious strtegies with vrious system prmeters on 16-node hypercube topology. I Legend Model:= Simultion: Dl I Time-units l e ~ 2.1!!! 1 8 % 1.5 S? $J 1.2 $.9.6 FRhdom Fhin FAv&ge PRhdom Phin PAve)ge FRndom FMin FAverge PRndom PMin PAverge Figure 62 =.5, PC = 16 tsk/time-uni~ Figure 7:1 =.8, #c = 16 tsk/time-unit, ps = 16 tsk/time-unig T= =.5 tim~units. A = 16 tsk/time-unit, Tu =.5 time-units. t3me-units Tree-units ] gy%) I (2.38%) (1.8%) (1.32%) FR&dom Fhin FAv;rge PR&dom Phin PAve;ge FRn om FMin FAverge PRndom PMin PAverge Figure 8: A =,8, PC = 16 tskk.ime-unit, Figure 9: A =.8, PC = 8 tsk/time-unit, /% = 8 tsldtime-unit, T,= 1. time-units. ps = 16 tsk/time-unit, T = 1. time-units. On the other hnd, FRndom nd PRndom yield identicl results by showing their insensitivity to the lod up dte method. In the second set, we hve selected 16-node ring network with medium system lod equl to,7. Agin, theresponse times predicted by the model mtch those produced by the simultion, s shown in Figure 11. Up to this point, the performnce of the model is compred with the sme simultion test cses through which empiricl dt for sttisticl modeling ws obtined. After chrcterizing PO nd E[NE], the queuing model ws used to compute the verge response time nd the results were compred with the sme simultion results. Therefore, the comprison of the model with simultion hs only reveled the correctness of the model. The vlidity of the proposed model is more strongly estblished s we obtin response time horn the model nd compre it with some dditionl 837

9 Comprison of response times predicted by the model nd simultion for vrious strtegies with vrious system prmeters on different network topologies. Legend: Model:- Time-units FRndom Ffiin FAverge PRklom PMin PAve>ge Figure 1 ~ =.6, PC = 12 tsk/time+nit, IZS = 12 tsk/time-unit, T. = 1.5 time-units. Topology = 16-node Folded Hypercube. Time-units I 1 FRndom Fkin FAv&ge PR&dom Phin PAv;rge Figure 11:2 =.7, PC =12 tsk/time-uni~ A = 12 tsk/time-unit, T. = 1. time-units. Topology = 16-node Ring.. Time-units FR&dom FMin FAv;rge PR&dom PMin PAve>ge Figure 121 =.7, AC = 16 tsk/time-unit, % = 16 tsk/time-unit, T. = 1. time-units. Topology = 9 node Mesh. FRtdom FMin FAverge PRndom PMin PAverge Figure 13:1 =.7, PC = 16 tsk/time-unit, AS = 16 tsk/time-unit, T. = 1. tim+units. Topology = 8 node Fully Conected. simultion runs. The empiricl dt from these simultion. P.4verge performs s well sfaverge, if TM is St_fIdl. runs hs not been used for sttisticl modeling. The ddi- All nerest neighbor lod blncing strtegies perform tionl simultion runs include different network topologies better if the number of links per node re incresed. This with different prmeters. The results of some combin- is becuse the probbility tht node finds suitble tions re shown in Figures 12 nd 13. By exmining these neighbor for tsk migrtion improves with the increse in figures, we conclude the following. the number of links.. Agtin, the difference bsvcen simultion nd model is smll. 838

10 TIW difference in the performnce of rndom strtegies nd rein strtegies is not very significnt scompred to the difference in the performnce of rndom nd verging strtegies. Rndom lgorithms cn be used with periodic informtion updtes for ny network topology becuse they generte less messge trffic. This is especilly true for the fully connected network where PRndom performs s well s FRndom.. If the ctul scheduling time, V for the rndom lgorithm is less thn tht for rein $ gonthms,prndom cn be used insted of FMin, PMin or FRndom. 5. Concluding Remrks In this pper, we hve presented n pproch for modeling the verge tsk response time for distributed lod blncing in multicomputer systems. With this pproch, we re ble to compre different lod brdncing schemes on unified bsis. We hve shown tht these strtegies cn be modeled by n open centrl server queuing network if the system is symmetric nd homogeneous. We believe tht ny sender-initited lod blncing strtegy cn be modeled by this queuing network. With exmples from wide rnge of system prmeters, it is shown tht the verge tsk response timepredictedthrough the proposed model closely mtches the response time obtined vi simultion. This pproch cn be useful for nlyzing nd tuning n existing system, nd evluting newly proposed strtegies. Acknowledgements The uthors express their grtitude to nonymous referees whose comments improved the qulity of this pper. Thnks re lso due to NPAC, Syrcuse University, in prticulr the system dministrtion stff, for mking Multimx vilble during the lst hours. References [1] I. Ahmd nd A. Ghfoor, A Semi Distributed Tsk Al- Ioction Strtegy for Lrge Hypercube Supercomput- ;;8~97Proc. of Supercomputing 9, Nov. 199, pp. [2] RymondM. Brynt nd Rphel A. Finkel, A Stble Distributed Scheduling Algorithm: in Proc. of 2nd Int 1. Conf on Distributed Computing Systems, 1981, pp [31 D. L. Eger, E. D. LzowskndJ. Zhorjn, Adptive Lod Shring in Homogeneous Distributed Systems, IEEE Trns. on Sofwre Erw.,vol. SE 12,. PD , My [4] K. Efe nd B. Groselj, Minimizing Control Overhed in A&ptive Lod Shring: in Proc. of 9 th Intl. Con. on Distributed Computing Syste~? 1989, pp [5] G. C. Fox, A. Kolw ndr. Wdhms, The Implementtion of Dynmic Lod Blncer, in Proc. of SIAM Hypercube A4uitiprocessors ConJ, 1987, pp [61 A. Ghfcor nd I. Ahmd. An Efficient Model of Dynmic Tsk Scheduling for Distributed Systems, in Proc. of COMPSAC 9, Oct.,199, pp [7] A. Ghfoor, T. Rshkow nd Imrn Ghfoor, Bisectionl Fult Tolernt Communiction Architecture for Supercomputer Systems, IEEE Trns. on Computers,vol. 38, no. 1, pp , October [8] D. C. Gnmdwld, B. A. Nzief ndd. A. Reed, Empiricl Comprison of Heuristic Lod Distribution in Point WPoint Multicomputer Networks: Proc. of The F~thDistributed Memory Computing Conference, April 199, pp [9] A. H cndt. J.Johnson, Sensitivity ShIdyof the Lod Blncing Algorithm in Distributed SystemYJourn/ of Prllel nd Distributed Computing, October 199, pp [1] L.V. KE, Compringtheperformnceof twodynmic lod distribution rneth6ds, Proceedings of Znt 1. Confi on Prllel Processing, 1988, pp [11] F. C. H. Lin nd R. M. Keller, Grdient Model: A demnd Driven Lod Blncing Scheme: in Proc. of d-th Int 1 Conf on Distributed Computing Systems, 1986, pp [121 M. L. nd M. Mehnn, Lod Blncing in Homoge- - neous Brodcst Distributed Systemsj h-proc. ofa~m Computer Network Pe@ormnce Symposium, April 1982, pp [13] L. M. Ni, C. Xu nd T. B. Gendreu, A Disrnbuted Drfting Algorithm for Lod Blncing, IEEE Trns. on Sof&re Eng., vol. SE-11, no. 1 Oct. 1985, pp [14] L. M. Ni nd Ki Hwng, Optiml Lod Blncing in Multiple Processor System with Mny Job Clsses, IEEE Trns. on Softwre Eng., vol. SE-11, My 1985, pp [15] K. Rmmrithm, J. A. Stnkovic nd Wei Zho, Distributed Scheduling of Tsks with Dedlines nd Resource Requirements, IEEE Trns. on Computers, vol. 38, no. 8, Aug. 1989, pp [16] A. Ross nd B. McMillin, Experimentl Comprison of Bidding nd Drfting Lod Shring Pmtocds, Proc. of The Fjlh Distributed Memory Computing Conference, April 199, pp [17] V. A. Sltore, A Distributed nd Adptive Dynmic Lod Blncing Scheme for Prllel Processing of Medium-Grin Tsks; Proc. of The Fifth Distributed ;9e.zoo9 Computing Conference, April 199, pp. [18] K. G. Sh~ nd Y. -C. Chng, LodShringin Distributed Rel-Time Systems with Stte-Chnge Brodcsts, IEEE Trns. on Computers, VOL 38, no. 8, Aug. 1989, pp [19] W. ShundL. V. K&5, ADynmic Scheduling Strtegy for the Chre-Kemel System, in Proc. of Supercomputing 89, November 1989, pp [2] J. A. Stnkovic nd I. S. Sidhu, An Adptive Bidding Algorithm for Processes, Clusters nd Distributed Groups, in Proc. of4 thint 1. Conf. on Distributed Computing Systems, 1984, pp [21] K. S. Trivedi, Probbility & Sttistics with Relibility, Queuing nd Computer Science Applictions, Prentice Hll, inc. Englewood Cliffs, NJ, [22] J. Xu nd K. Hwng, Heuristic Methods for Dynmic Lod Brdncing in Messge-Pssing Supercomputer, in Proc. of Supercomputing 9, November 199, pp. 88U397. [23] Y. Wng ndr. J. T. Morris, Lod Shring in Distributed Systems, inieee Trns. on Computers, C 34 no. 3, Mrch 1985, pp

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